2isqrt (example 3.6)

Percentage Accurate: 68.5% → 99.7%

Time: 9.4s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(1 + x\right)}^{-0.5}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, -t_0, {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (+ 1.0 x) -0.5))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-6)
     (*
      (-
       (+ (/ 0.5 x) (/ 0.3125 (pow x 3.0)))
       (+ (/ 0.375 (* x x)) (/ 0.2734375 (pow x 4.0))))
      (pow x -0.5))
     (fma t_0 (- t_0) (pow x -0.5)))))

C

double code(double x) {
	double t_0 = sqrt(pow((1.0 + x), -0.5));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-6) {
		tmp = (((0.5 / x) + (0.3125 / pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / pow(x, 4.0)))) * pow(x, -0.5);
	} else {
		tmp = fma(t_0, -t_0, pow(x, -0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt((Float64(1.0 + x) ^ -0.5))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 / x) + Float64(0.3125 / (x ^ 3.0))) - Float64(Float64(0.375 / Float64(x * x)) + Float64(0.2734375 / (x ^ 4.0)))) * (x ^ -0.5));
	else
		tmp = fma(t_0, Float64(-t_0), (x ^ -0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(N[(0.5 / x), $MachinePrecision] + N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.2734375 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * (-t$95$0) + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999991e-6

   1. Initial program 35.6%

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

      1. frac-sub 35.7%

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

      2. div-inv 35.7%

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

      3. *-un-lft-identity 35.7%

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

      4. +-commutative 35.7%

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

      5. *-rgt-identity 35.7%

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

      6. metadata-eval 35.7%

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

      7. frac-times 35.7%

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

      8. un-div-inv 35.7%

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

      9. pow1/2 35.7%

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

      10. pow-flip 35.7%

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

      11. metadata-eval 35.7%

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

      12. +-commutative 35.7%

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

   3. Applied egg-rr 35.7%

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

      1. associate-*r/ 35.7%

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

      2. *-rgt-identity 35.7%

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

      3. times-frac 35.7%

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

      4. div-sub 35.6%

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

      5. *-inverses 35.6%

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

      6. /-rgt-identity 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{\color{blue}{0.5}}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.3125}}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      5. +-commutative 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \color{blue}{\left(0.375 \cdot \frac{1}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)}\right) \cdot {x}^{-0.5} \]

      6. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      7. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{\color{blue}{0.375}}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      8. unpow2 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{\color{blue}{x \cdot x}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      9. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \color{blue}{\frac{0.2734375 \cdot 1}{{x}^{4}}}\right)\right) \cdot {x}^{-0.5} \]

      10. metadata-eval 99.6%

          \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{\color{blue}{0.2734375}}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)} \cdot {x}^{-0.5} \]

   if 1.99999999999999991e-6 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. fma-def 99.6%

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

      6. inv-pow 99.6%

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

      7. sqrt-pow2 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. inv-pow 99.6%

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

      11. sqrt-pow2 99.6%

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

      12. +-commutative 99.6%

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

      13. metadata-eval 99.6%

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

      14. pow1/2 99.6%

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

      15. pow-flip 100.0%

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

      16. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{\left(1 + x\right)}^{-0.5}}, -\sqrt{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{\left(1 + x\right)}^{-0.5}}, -\sqrt{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-6)
   (*
    (-
     (+ (/ 0.5 x) (/ 0.3125 (pow x 3.0)))
     (+ (/ 0.375 (* x x)) (/ 0.2734375 (pow x 4.0))))
    (pow x -0.5))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-6) {
		tmp = (((0.5 / x) + (0.3125 / pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / pow(x, 4.0)))) * pow(x, -0.5);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-6) then
        tmp = (((0.5d0 / x) + (0.3125d0 / (x ** 3.0d0))) - ((0.375d0 / (x * x)) + (0.2734375d0 / (x ** 4.0d0)))) * (x ** (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-6) {
		tmp = (((0.5 / x) + (0.3125 / Math.pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / Math.pow(x, 4.0)))) * Math.pow(x, -0.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-6:
		tmp = (((0.5 / x) + (0.3125 / math.pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / math.pow(x, 4.0)))) * math.pow(x, -0.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 / x) + Float64(0.3125 / (x ^ 3.0))) - Float64(Float64(0.375 / Float64(x * x)) + Float64(0.2734375 / (x ^ 4.0)))) * (x ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-6)
		tmp = (((0.5 / x) + (0.3125 / (x ^ 3.0))) - ((0.375 / (x * x)) + (0.2734375 / (x ^ 4.0)))) * (x ^ -0.5);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(N[(0.5 / x), $MachinePrecision] + N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.2734375 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999991e-6

   1. Initial program 35.6%

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

      1. frac-sub 35.7%

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

      2. div-inv 35.7%

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

      3. *-un-lft-identity 35.7%

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

      4. +-commutative 35.7%

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

      5. *-rgt-identity 35.7%

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

      6. metadata-eval 35.7%

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

      7. frac-times 35.7%

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

      8. un-div-inv 35.7%

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

      9. pow1/2 35.7%

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

      10. pow-flip 35.7%

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

      11. metadata-eval 35.7%

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

      12. +-commutative 35.7%

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

   3. Applied egg-rr 35.7%

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

      1. associate-*r/ 35.7%

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

      2. *-rgt-identity 35.7%

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

      3. times-frac 35.7%

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

      4. div-sub 35.6%

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

      5. *-inverses 35.6%

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

      6. /-rgt-identity 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{\color{blue}{0.5}}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.3125}}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      5. +-commutative 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \color{blue}{\left(0.375 \cdot \frac{1}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)}\right) \cdot {x}^{-0.5} \]

      6. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      7. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{\color{blue}{0.375}}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      8. unpow2 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{\color{blue}{x \cdot x}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

      9. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \color{blue}{\frac{0.2734375 \cdot 1}{{x}^{4}}}\right)\right) \cdot {x}^{-0.5} \]

      10. metadata-eval 99.6%

          \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{\color{blue}{0.2734375}}{{x}^{4}}\right)\right) \cdot {x}^{-0.5} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)} \cdot {x}^{-0.5} \]

   if 1.99999999999999991e-6 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]

      4. mul0-lft 100.0%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-8)
   (* (pow x -0.5) (- (+ (/ 0.5 x) (/ 0.3125 (pow x 3.0))) (/ 0.375 (* x x))))
   (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8) {
		tmp = pow(x, -0.5) * (((0.5 / x) + (0.3125 / pow(x, 3.0))) - (0.375 / (x * x)));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-8) then
        tmp = (x ** (-0.5d0)) * (((0.5d0 / x) + (0.3125d0 / (x ** 3.0d0))) - (0.375d0 / (x * x)))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-8) {
		tmp = Math.pow(x, -0.5) * (((0.5 / x) + (0.3125 / Math.pow(x, 3.0))) - (0.375 / (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-8:
		tmp = math.pow(x, -0.5) * (((0.5 / x) + (0.3125 / math.pow(x, 3.0))) - (0.375 / (x * x)))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-8)
		tmp = Float64((x ^ -0.5) * Float64(Float64(Float64(0.5 / x) + Float64(0.3125 / (x ^ 3.0))) - Float64(0.375 / Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8)
		tmp = (x ^ -0.5) * (((0.5 / x) + (0.3125 / (x ^ 3.0))) - (0.375 / (x * x)));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(N[(0.5 / x), $MachinePrecision] + N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2e-8

   1. Initial program 35.2%

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

      1. frac-sub 35.3%

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

      2. div-inv 35.3%

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

      3. *-un-lft-identity 35.3%

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

      4. +-commutative 35.3%

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

      5. *-rgt-identity 35.3%

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

      6. metadata-eval 35.3%

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

      7. frac-times 35.3%

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

      8. un-div-inv 35.3%

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

      9. pow1/2 35.3%

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

      10. pow-flip 35.3%

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

      11. metadata-eval 35.3%

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

      12. +-commutative 35.3%

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

   3. Applied egg-rr 35.3%

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

      1. associate-*r/ 35.3%

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

      2. *-rgt-identity 35.3%

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

      3. times-frac 35.3%

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

      4. div-sub 35.3%

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

      5. *-inverses 35.3%

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

      6. /-rgt-identity 35.3%

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

   5. Simplified 35.3%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{\color{blue}{0.5}}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.3125}}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. associate-*r/ 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      6. metadata-eval 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      7. unpow2 99.6%

         \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]

   if 2e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. frac-sub 99.5%

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

      2. div-inv 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. metadata-eval 99.5%

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

      7. frac-times 99.5%

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

      8. un-div-inv 99.5%

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

      9. pow1/2 99.5%

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

      10. pow-flip 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. times-frac 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. /-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. sqrt-undiv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

      \[\leadsto \left(1 - \color{blue}{\sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\left(\frac{0.5}{x} + \frac{0.3125}{{x}^{3}}\right) - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-11)
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))
   (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-11) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-11) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-11) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-11:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-11)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-11)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999988e-11

   1. Initial program 34.6%

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

      1. frac-sub 34.7%

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

      2. div-inv 34.7%

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

      3. *-un-lft-identity 34.7%

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

      4. +-commutative 34.7%

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

      5. *-rgt-identity 34.7%

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

      6. metadata-eval 34.7%

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

      7. frac-times 34.7%

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

      8. un-div-inv 34.7%

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

      9. pow1/2 34.7%

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

      10. pow-flip 34.7%

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

      11. metadata-eval 34.7%

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

      12. +-commutative 34.7%

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

   3. Applied egg-rr 34.7%

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

      1. associate-*r/ 34.7%

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

      2. *-rgt-identity 34.7%

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

      3. times-frac 34.7%

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

      4. div-sub 34.7%

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

      5. *-inverses 34.7%

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

      6. /-rgt-identity 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow2 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]

   if 1.99999999999999988e-11 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.1%

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

      1. frac-sub 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. +-commutative 99.1%

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

      5. *-rgt-identity 99.1%

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

      6. metadata-eval 99.1%

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

      7. frac-times 99.1%

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

      8. un-div-inv 99.1%

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

      9. pow1/2 99.1%

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

      10. pow-flip 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.5%

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

      4. div-sub 99.4%

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

      5. *-inverses 99.4%

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

      6. /-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. sqrt-undiv 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

   9. Simplified 99.4%

      \[\leadsto \left(1 - \color{blue}{\sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-11)
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-11) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-11) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-11) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-11:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-11)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-11)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999988e-11

   1. Initial program 34.6%

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

      1. frac-sub 34.7%

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

      2. div-inv 34.7%

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

      3. *-un-lft-identity 34.7%

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

      4. +-commutative 34.7%

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

      5. *-rgt-identity 34.7%

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

      6. metadata-eval 34.7%

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

      7. frac-times 34.7%

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

      8. un-div-inv 34.7%

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

      9. pow1/2 34.7%

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

      10. pow-flip 34.7%

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

      11. metadata-eval 34.7%

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

      12. +-commutative 34.7%

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

   3. Applied egg-rr 34.7%

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

      1. associate-*r/ 34.7%

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

      2. *-rgt-identity 34.7%

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

      3. times-frac 34.7%

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

      4. div-sub 34.7%

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

      5. *-inverses 34.7%

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

      6. /-rgt-identity 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow2 99.6%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]

   if 1.99999999999999988e-11 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. clear-num 99.1%

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

      3. associate-/r/ 99.1%

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

      4. prod-diff 99.1%

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

      5. *-un-lft-identity 99.1%

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

      6. fma-neg 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. inv-pow 99.1%

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

      9. sqrt-pow2 99.4%

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

      10. metadata-eval 99.4%

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

      11. pow1/2 99.4%

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

      12. pow-flip 99.4%

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

      13. +-commutative 99.4%

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

      14. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]

      4. mul0-lft 99.4%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]

      5. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+ (pow x -0.5) (- -1.0 (* x -0.5)))
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.1:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * -0.5))
	else:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * -0.5)));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.1)
		tmp = (x ^ -0.5) + (-1.0 - (x * -0.5));
	else
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 98.8%

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

      6. pow1/2 98.8%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - 1 \]

      7. pow-flip 99.2%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - 1 \]

      8. metadata-eval 99.2%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - 1 \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \left(-0.5 \cdot x + 1\right) \]
   5. Step-by-step derivation

      1. +-lft-identity 99.2%

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

   6. Simplified 99.9%

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

   if 1.1000000000000001 < x

   1. Initial program 36.1%

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

      1. frac-sub 36.2%

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

      2. div-inv 36.2%

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

      3. *-un-lft-identity 36.2%

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

      4. +-commutative 36.2%

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

      5. *-rgt-identity 36.2%

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

      6. metadata-eval 36.2%

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

      7. frac-times 36.2%

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

      8. un-div-inv 36.2%

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

      9. pow1/2 36.2%

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

      10. pow-flip 36.2%

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

      11. metadata-eval 36.2%

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

      12. +-commutative 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. associate-*r/ 36.2%

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

      2. *-rgt-identity 36.2%

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

      3. times-frac 36.2%

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

      4. div-sub 36.1%

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

      5. *-inverses 36.1%

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

      6. /-rgt-identity 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.2%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval 98.2%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/ 98.2%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval 98.2%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow2 98.2%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 98.8%

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

      6. pow1/2 98.8%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - 1 \]

      7. pow-flip 99.2%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - 1 \]

      8. metadata-eval 99.2%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - 1 \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \left(-0.5 \cdot x + 1\right) \]
   5. Step-by-step derivation

      1. +-lft-identity 99.2%

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

   6. Simplified 99.9%

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

   if 1 < x

   1. Initial program 36.1%

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

      1. frac-sub 36.2%

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

      2. div-inv 36.2%

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

      3. *-un-lft-identity 36.2%

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

      4. +-commutative 36.2%

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

      5. *-rgt-identity 36.2%

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

      6. metadata-eval 36.2%

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

      7. frac-times 36.2%

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

      8. un-div-inv 36.2%

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

      9. pow1/2 36.2%

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

      10. pow-flip 36.2%

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

      11. metadata-eval 36.2%

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

      12. +-commutative 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. associate-*r/ 36.2%

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

      2. *-rgt-identity 36.2%

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

      3. times-frac 36.2%

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

      4. div-sub 36.1%

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

      5. *-inverses 36.1%

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

      6. /-rgt-identity 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in x around inf 97.0%

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

4. Final simplification 98.4%

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

Alternative 8: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* (/ 0.5 x) (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.8%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 98.8%

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

      6. pow1/2 98.8%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - 1 \]

      7. pow-flip 99.2%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - 1 \]

      8. metadata-eval 99.2%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - 1 \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - 1 \]
   5. Step-by-step derivation

      1. +-lft-identity 99.2%

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

   6. Simplified 99.2%

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

   if 0.680000000000000049 < x

   1. Initial program 36.1%

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

      1. frac-sub 36.2%

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

      2. div-inv 36.2%

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

      3. *-un-lft-identity 36.2%

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

      4. +-commutative 36.2%

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

      5. *-rgt-identity 36.2%

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

      6. metadata-eval 36.2%

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

      7. frac-times 36.2%

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

      8. un-div-inv 36.2%

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

      9. pow1/2 36.2%

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

      10. pow-flip 36.2%

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

      11. metadata-eval 36.2%

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

      12. +-commutative 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. associate-*r/ 36.2%

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

      2. *-rgt-identity 36.2%

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

      3. times-frac 36.2%

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

      4. div-sub 36.1%

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

      5. *-inverses 36.1%

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

      6. /-rgt-identity 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in x around inf 97.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 9: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.81:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.81) (+ (pow x -0.5) -1.0) (pow x -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.81) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.81d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = x ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.81) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.81:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.81)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = x ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.81)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = x ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.81], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[x, -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.81000000000000005

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.8%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 98.8%

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

      6. pow1/2 98.8%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - 1 \]

      7. pow-flip 99.2%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - 1 \]

      8. metadata-eval 99.2%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - 1 \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - 1 \]
   5. Step-by-step derivation

      1. +-lft-identity 99.2%

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

   6. Simplified 99.2%

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

   if 0.81000000000000005 < x

   1. Initial program 36.1%

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

      1. frac-sub 36.2%

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

      2. div-inv 36.2%

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

      3. *-un-lft-identity 36.2%

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

      4. +-commutative 36.2%

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

      5. *-rgt-identity 36.2%

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

      6. metadata-eval 36.2%

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

      7. frac-times 36.2%

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

      8. un-div-inv 36.2%

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

      9. pow1/2 36.2%

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

      10. pow-flip 36.2%

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

      11. metadata-eval 36.2%

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

      12. +-commutative 36.2%

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

   3. Applied egg-rr 36.2%

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

      1. associate-*r/ 36.2%

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

      2. *-rgt-identity 36.2%

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

      3. times-frac 36.2%

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

      4. div-sub 36.1%

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

      5. *-inverses 36.1%

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

      6. /-rgt-identity 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in x around 0 5.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.81:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \]

Alternative 10: 50.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -0.5))

C

double code(double x) {
	return pow(x, -0.5);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5)

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = x ^ -0.5;
end

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 67.3%

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

   1. frac-sub 67.4%

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

   2. div-inv 67.4%

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

   3. *-un-lft-identity 67.4%

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

   4. +-commutative 67.4%

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

   5. *-rgt-identity 67.4%

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

   6. metadata-eval 67.4%

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

   7. frac-times 67.4%

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

   8. un-div-inv 67.4%

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

   9. pow1/2 67.4%

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

   10. pow-flip 67.6%

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

   11. metadata-eval 67.6%

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

   12. +-commutative 67.6%

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

3. Applied egg-rr 67.6%

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

   1. associate-*r/ 67.6%

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

   2. *-rgt-identity 67.6%

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

   3. times-frac 67.6%

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

   4. div-sub 67.6%

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

   5. *-inverses 67.6%

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

   6. /-rgt-identity 67.6%

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

5. Simplified 67.6%

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

6. Taylor expanded in x around 0 49.1%

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

7. Final simplification 49.1%

   \[\leadsto {x}^{-0.5} \]

Alternative 11: 3.9% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0 50.8%

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

3. Taylor expanded in x around inf 4.1%

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

4. Final simplification 4.1%

   \[\leadsto x \cdot 0.5 \]

Developer target: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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