2isqrt (example 3.6)

Percentage Accurate: 69.4% → 99.7%

Time: 12.2s

Alternatives: 13

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: 69.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (* (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (* x x)))) (pow x -0.5))
   (* (pow x -0.5) (- 1.0 (/ (sqrt x) (pow (pow (+ 1.0 x) 0.25) 2.0))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / (x * x)))) * pow(x, -0.5);
	} else {
		tmp = pow(x, -0.5) * (1.0 - (sqrt(x) / pow(pow((1.0 + x), 0.25), 2.0)));
	}
	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)))) <= 1d-9) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) + ((-0.375d0) / (x * x)))) * (x ** (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - (sqrt(x) / (((1.0d0 + x) ** 0.25d0) ** 2.0d0)))
    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)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / (x * x)))) * Math.pow(x, -0.5);
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - (Math.sqrt(x) / Math.pow(Math.pow((1.0 + x), 0.25), 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 1e-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / (x * x)))) * math.pow(x, -0.5)
	else:
		tmp = math.pow(x, -0.5) * (1.0 - (math.sqrt(x) / math.pow(math.pow((1.0 + x), 0.25), 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 1e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / Float64(x * x)))) * (x ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - Float64(sqrt(x) / ((Float64(1.0 + x) ^ 0.25) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-9)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) + (-0.375 / (x * x)))) * (x ^ -0.5);
	else
		tmp = (x ^ -0.5) * (1.0 - (sqrt(x) / (((1.0 + x) ^ 0.25) ^ 2.0)));
	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], 1e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] / N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $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)))) < 1.00000000000000006e-9

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. +-commutative 99.6%

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

       3. associate-+l+ 99.6%

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

       4. associate-*r/ 99.6%

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

       5. metadata-eval 99.6%

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

       6. associate-*r/ 99.6%

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

       7. metadata-eval 99.6%

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

       8. associate-*r/ 99.6%

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

       9. metadata-eval 99.6%

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

       10. distribute-neg-frac 99.6%

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

       11. metadata-eval 99.6%

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

       12. unpow2 99.6%

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

   12. Simplified 99.6%

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

   if 1.00000000000000006e-9 < (-.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. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.6%

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

      8. un-div-inv 99.6%

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

      9. pow1/2 99.6%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. metadata-eval 100.0%

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

      9. exp-to-pow 100.0%

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

      10. metadata-eval 100.0%

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

      11. exp-to-pow 100.0%

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

      12. hypot-1-def 100.0%

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

      13. exp-to-pow 100.0%

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

      14. unpow1/2 100.0%

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

      15. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. hypot-1-def 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-to-exp 100.0%

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

      5. log1p-udef 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. pow2 100.0%

         \[\leadsto \left(1 - \frac{\sqrt{x}}{\color{blue}{{\left(\sqrt{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right)}^{2}}}\right) \cdot {x}^{-0.5} \]

      8. log1p-udef 100.0%

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

      9. pow-to-exp 100.0%

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

      10. sqrt-pow1 100.0%

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

      11. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \left(1 - \frac{\sqrt{x}}{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}}\right) \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{x} \cdot {\left(1 + x\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (* (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (* x x)))) (pow x -0.5))
   (* (pow x -0.5) (- 1.0 (* (sqrt x) (pow (+ 1.0 x) -0.5))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / (x * x)))) * pow(x, -0.5);
	} else {
		tmp = pow(x, -0.5) * (1.0 - (sqrt(x) * 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)))) <= 1d-9) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) + ((-0.375d0) / (x * x)))) * (x ** (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - (sqrt(x) * ((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)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / (x * x)))) * Math.pow(x, -0.5);
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - (Math.sqrt(x) * 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)))) <= 1e-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / (x * x)))) * math.pow(x, -0.5)
	else:
		tmp = math.pow(x, -0.5) * (1.0 - (math.sqrt(x) * 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)))) <= 1e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / Float64(x * x)))) * (x ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - Float64(sqrt(x) * (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)))) <= 1e-9)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) + (-0.375 / (x * x)))) * (x ^ -0.5);
	else
		tmp = (x ^ -0.5) * (1.0 - (sqrt(x) * ((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], 1e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] * N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $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)))) < 1.00000000000000006e-9

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. +-commutative 99.6%

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

       3. associate-+l+ 99.6%

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

       4. associate-*r/ 99.6%

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

       5. metadata-eval 99.6%

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

       6. associate-*r/ 99.6%

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

       7. metadata-eval 99.6%

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

       8. associate-*r/ 99.6%

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

       9. metadata-eval 99.6%

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

       10. distribute-neg-frac 99.6%

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

       11. metadata-eval 99.6%

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

       12. unpow2 99.6%

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

   12. Simplified 99.6%

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

   if 1.00000000000000006e-9 < (-.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. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.6%

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

      8. un-div-inv 99.6%

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

      9. pow1/2 99.6%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. metadata-eval 100.0%

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

      9. exp-to-pow 100.0%

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

      10. metadata-eval 100.0%

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

      11. exp-to-pow 100.0%

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

      12. hypot-1-def 100.0%

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

      13. exp-to-pow 100.0%

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

      14. unpow1/2 100.0%

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

      15. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. hypot-1-def 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-to-exp 100.0%

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

      5. log1p-udef 100.0%

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

      6. div-inv 100.0%

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

      7. log1p-udef 100.0%

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

      8. pow-to-exp 100.0%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{x} \cdot {\left(1 + x\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (* (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (* x x)))) (pow x -0.5))
   (* (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)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / (x * x)))) * pow(x, -0.5);
	} 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)))) <= 1d-9) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) + ((-0.375d0) / (x * x)))) * (x ** (-0.5d0))
    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)))) <= 1e-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / (x * x)))) * Math.pow(x, -0.5);
	} 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)))) <= 1e-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / (x * x)))) * math.pow(x, -0.5)
	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)))) <= 1e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / Float64(x * x)))) * (x ^ -0.5));
	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)))) <= 1e-9)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) + (-0.375 / (x * x)))) * (x ^ -0.5);
	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], 1e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $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)))) < 1.00000000000000006e-9

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. +-commutative 99.6%

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

       3. associate-+l+ 99.6%

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

       4. associate-*r/ 99.6%

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

       5. metadata-eval 99.6%

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

       6. associate-*r/ 99.6%

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

       7. metadata-eval 99.6%

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

       8. associate-*r/ 99.6%

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

       9. metadata-eval 99.6%

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

       10. distribute-neg-frac 99.6%

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

       11. metadata-eval 99.6%

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

       12. unpow2 99.6%

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

   12. Simplified 99.6%

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

   if 1.00000000000000006e-9 < (-.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. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.6%

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

      8. un-div-inv 99.6%

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

      9. pow1/2 99.6%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. metadata-eval 100.0%

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

      9. exp-to-pow 100.0%

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

      10. metadata-eval 100.0%

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

      11. exp-to-pow 100.0%

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

      12. hypot-1-def 100.0%

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

      13. exp-to-pow 100.0%

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

      14. unpow1/2 100.0%

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

      15. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. hypot-1-def 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-undiv 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (* (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)))) <= 1e-9) {
		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)))) <= 1d-9) 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)))) <= 1e-9) {
		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)))) <= 1e-9:
		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)))) <= 1e-9)
		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)))) <= 1e-9)
		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], 1e-9], 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)))) < 1.00000000000000006e-9

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow2 99.5%

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

   12. Simplified 99.5%

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

   if 1.00000000000000006e-9 < (-.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. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.6%

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

      8. un-div-inv 99.6%

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

      9. pow1/2 99.6%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. metadata-eval 100.0%

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

      9. exp-to-pow 100.0%

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

      10. metadata-eval 100.0%

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

      11. exp-to-pow 100.0%

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

      12. hypot-1-def 100.0%

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

      13. exp-to-pow 100.0%

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

      14. unpow1/2 100.0%

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

      15. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. hypot-1-def 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-undiv 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (* (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)))) <= 1e-9) {
		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)))) <= 1d-9) 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)))) <= 1e-9) {
		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)))) <= 1e-9:
		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)))) <= 1e-9)
		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)))) <= 1e-9)
		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], 1e-9], 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)))) < 1.00000000000000006e-9

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow2 99.5%

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

   12. Simplified 99.5%

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

   if 1.00000000000000006e-9 < (-.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. neg-mul-1 100.0%

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

      3. rem-log-exp 100.0%

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

      4. log-rec 100.0%

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

      5. +-commutative 100.0%

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

      6. log-rec 100.0%

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

      7. rem-log-exp 100.0%

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

      8. sub-neg 100.0%

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

      9. +-inverses 100.0%

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

      10. +-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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (* (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) + (x * 0.5)) + -1.0;
	} 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)) + (x * 0.5d0)) + (-1.0d0)
    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) + (x * 0.5)) + -1.0;
	} 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) + (x * 0.5)) + -1.0
	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(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	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) + (x * 0.5)) + -1.0;
	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[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $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. 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-cube-cbrt 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]

   4. Taylor expanded in x around 0 97.3%

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

   if 1.1000000000000001 < x

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow2 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 98.3%

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

Alternative 7: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (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 <= 1.0d0) then
        tmp = ((x ** (-0.5d0)) + (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 <= 1.0) {
		tmp = (Math.pow(x, -0.5) + (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 <= 1.0:
		tmp = (math.pow(x, -0.5) + (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 <= 1.0)
		tmp = Float64(Float64((x ^ -0.5) + 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 <= 1.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (0.5 / x) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $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 < 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-cube-cbrt 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]

   4. Taylor expanded in x around 0 97.3%

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

   if 1 < x

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 97.7%

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

Alternative 8: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.66) (+ (pow x -0.5) -1.0) (* (/ 0.5 x) (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		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 <= 0.66d0) 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 <= 0.66) {
		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 <= 0.66:
		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 <= 0.66)
		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 <= 0.66)
		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, 0.66], 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 < 0.660000000000000031

   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-cube-cbrt 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]

   4. Taylor expanded in x around 0 96.9%

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

   if 0.660000000000000031 < x

   1. Initial program 40.9%

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

      1. frac-sub 41.0%

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

      2. div-inv 41.0%

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

      3. *-un-lft-identity 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

      6. metadata-eval 41.0%

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

      7. frac-times 41.0%

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

      8. un-div-inv 41.0%

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

      9. pow1/2 41.0%

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

      10. pow-flip 41.0%

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

      11. metadata-eval 41.0%

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

      12. +-commutative 41.0%

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

   3. Applied egg-rr 41.0%

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

      1. associate-*r/ 41.0%

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

      2. *-rgt-identity 41.0%

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

      3. times-frac 41.0%

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

      4. div-sub 41.0%

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

      5. *-inverses 41.0%

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

      6. unpow1 41.0%

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

      7. sqr-pow 41.0%

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

      8. metadata-eval 41.0%

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

      9. exp-to-pow 6.7%

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

      10. metadata-eval 6.7%

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

      11. exp-to-pow 6.6%

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

      12. hypot-1-def 6.5%

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

      13. exp-to-pow 41.0%

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

      14. unpow1/2 41.0%

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

      15. /-rgt-identity 41.0%

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

   5. Simplified 41.0%

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

      1. hypot-1-def 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. pow1/2 22.1%

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

      5. pow-to-exp 6.5%

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

      6. log1p-udef 6.5%

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

      7. *-un-lft-identity 6.5%

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

      8. times-frac 6.5%

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

      9. pow1/2 6.5%

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

      10. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      11. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{x}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      12. pow1/2 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{x}^{0.5}}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      13. sqrt-pow1 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      14. metadata-eval 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{\color{blue}{0.25}}}{e^{\mathsf{log1p}\left(x\right) \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      15. log1p-udef 6.6%

          \[\leadsto \left(1 - \frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.5}}\right) \cdot {x}^{-0.5} \]

      16. pow-to-exp 27.4%

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

      17. pow1/2 27.5%

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

   7. Applied egg-rr 27.5%

      \[\leadsto \left(1 - \color{blue}{\frac{{x}^{0.25}}{1} \cdot \frac{{x}^{0.25}}{\sqrt{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 22.0%

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

      3. pow-sqr 41.0%

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

      4. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 9: 53.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. pow1/2 71.4%

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

   2. pow-to-exp 54.9%

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

   3. +-commutative 54.9%

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

   4. log1p-udef 54.9%

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

3. Applied egg-rr 54.9%

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

5. Applied egg-rr 52.3%

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

6. Taylor expanded in x around 0 53.3%

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

   1. distribute-rgt-in 53.3%

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

   2. *-lft-identity 53.3%

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

   3. pow-plus 53.6%

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

   4. metadata-eval 53.6%

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

   5. unpow1/2 53.6%

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

8. Simplified 53.6%

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

9. Final simplification 53.6%

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

Alternative 10: 51.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. pow1/2 71.4%

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

   2. pow-to-exp 54.9%

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

   3. +-commutative 54.9%

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

   4. log1p-udef 54.9%

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

3. Applied egg-rr 54.9%

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

5. Applied egg-rr 52.3%

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

6. Taylor expanded in x around inf 51.6%

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

7. Final simplification 51.6%

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

Alternative 11: 51.3% 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 71.4%

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

   1. frac-sub 71.4%

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

   2. div-inv 71.4%

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

   3. *-un-lft-identity 71.4%

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

   4. +-commutative 71.4%

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

   5. *-rgt-identity 71.4%

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

   6. metadata-eval 71.4%

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

   7. frac-times 71.4%

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

   8. un-div-inv 71.4%

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

   9. pow1/2 71.4%

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

   10. pow-flip 71.6%

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

   11. metadata-eval 71.6%

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

   12. +-commutative 71.6%

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

3. Applied egg-rr 71.6%

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

   1. associate-*r/ 71.6%

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

   2. *-rgt-identity 71.6%

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

   3. times-frac 71.6%

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

   4. div-sub 71.7%

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

   5. *-inverses 71.7%

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

   6. unpow1 71.7%

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

   7. sqr-pow 71.6%

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

   8. metadata-eval 71.6%

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

   9. exp-to-pow 55.2%

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

   10. metadata-eval 55.2%

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

   11. exp-to-pow 55.1%

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

   12. hypot-1-def 55.1%

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

   13. exp-to-pow 71.6%

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

   14. unpow1/2 71.7%

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

   15. /-rgt-identity 71.7%

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

5. Simplified 71.7%

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

6. Taylor expanded in x around 0 51.8%

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

7. Final simplification 51.8%

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

Alternative 12: 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 71.4%

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

   1. pow1/2 71.4%

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

   2. pow-to-exp 54.9%

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

   3. +-commutative 54.9%

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

   4. log1p-udef 54.9%

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

3. Applied egg-rr 54.9%

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

4. Taylor expanded in x around 0 51.9%

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

   1. *-commutative 51.9%

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

6. Simplified 51.9%

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

7. Taylor expanded in x around inf 4.0%

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

8. Final simplification 4.0%

   \[\leadsto x \cdot 0.5 \]

Alternative 13: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 71.4%

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

2. Taylor expanded in x around 0 51.4%

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

3. Taylor expanded in x around inf 1.9%

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

4. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 99.1% 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 2023279 
(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)))))