2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.5%

Time: 18.9s

Alternatives: 9

Speedup: 1.9×

Specification

\[x > 1 \land x < 10^{+308}\]

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: 38.4% 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: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{t\_0}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))))
   (if (<= x 2e+116) (/ (/ (/ 1.0 x) (+ 1.0 x)) t_0) (/ (/ (/ 1.0 t_0) x) x))))

C

double code(double x) {
	double t_0 = pow(x, -0.5) + pow((1.0 + x), -0.5);
	double tmp;
	if (x <= 2e+116) {
		tmp = ((1.0 / x) / (1.0 + x)) / t_0;
	} else {
		tmp = ((1.0 / t_0) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))
    if (x <= 2d+116) then
        tmp = ((1.0d0 / x) / (1.0d0 + x)) / t_0
    else
        tmp = ((1.0d0 / t_0) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5);
	double tmp;
	if (x <= 2e+116) {
		tmp = ((1.0 / x) / (1.0 + x)) / t_0;
	} else {
		tmp = ((1.0 / t_0) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)
	tmp = 0
	if x <= 2e+116:
		tmp = ((1.0 / x) / (1.0 + x)) / t_0
	else:
		tmp = ((1.0 / t_0) / x) / x
	return tmp

Julia

function code(x)
	t_0 = Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))
	tmp = 0.0
	if (x <= 2e+116)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(1.0 + x)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / t_0) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x ^ -0.5) + ((1.0 + x) ^ -0.5);
	tmp = 0.0;
	if (x <= 2e+116)
		tmp = ((1.0 / x) / (1.0 + x)) / t_0;
	else
		tmp = ((1.0 / t_0) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+116], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000003e116

   1. Initial program 12.2%

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

      1. flip-- 12.1%

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

      2. div-inv 12.1%

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

      3. frac-times 12.3%

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

      4. metadata-eval 12.3%

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

      5. add-sqr-sqrt 12.5%

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

      6. frac-times 12.6%

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

      7. metadata-eval 12.6%

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

      8. add-sqr-sqrt 12.5%

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

      9. +-commutative 12.5%

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

      10. inv-pow 12.5%

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

      11. sqrt-pow2 12.5%

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

      12. metadata-eval 12.5%

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

      13. pow1/2 12.5%

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

      14. pow-flip 12.5%

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

      15. +-commutative 12.5%

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

      16. metadata-eval 12.5%

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

   4. Applied egg-rr 12.5%

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

      1. *-commutative 12.5%

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

      2. frac-sub 16.8%

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

      3. frac-times 16.8%

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

      4. *-un-lft-identity 16.8%

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

      5. *-un-lft-identity 16.8%

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

      6. *-rgt-identity 16.8%

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

      7. distribute-lft-in 16.8%

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

      8. *-rgt-identity 16.8%

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

      9. pow2 16.8%

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

   6. Applied egg-rr 16.8%

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

      1. *-commutative 16.8%

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

      2. associate-/r* 16.8%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. unpow2 99.5%

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

      8. fma-define 99.6%

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

   8. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. fma-undefine 99.5%

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

      3. *-rgt-identity 99.5%

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

      4. distribute-lft-in 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 99.3%

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

      7. inv-pow 99.3%

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

      8. metadata-eval 99.3%

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

      9. pow-prod-up 99.3%

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

      10. associate-*l* 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r* 99.3%

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

       2. pow-sqr 99.3%

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

       3. metadata-eval 99.3%

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

       4. unpow-1 99.3%

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

       5. associate-*r/ 99.5%

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

       6. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 2.00000000000000003e116 < x

   1. Initial program 53.2%

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

      1. flip-- 53.2%

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

      2. div-inv 53.2%

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

      3. frac-times 30.0%

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

      4. metadata-eval 30.0%

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

      5. add-sqr-sqrt 24.7%

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

      6. frac-times 35.5%

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

      7. metadata-eval 35.5%

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

      8. add-sqr-sqrt 53.2%

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

      9. +-commutative 53.2%

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

      10. inv-pow 53.2%

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

      11. sqrt-pow2 53.2%

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

      12. metadata-eval 53.2%

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

      13. pow1/2 53.2%

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

      14. pow-flip 53.2%

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

      15. +-commutative 53.2%

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

      16. metadata-eval 53.2%

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

   4. Applied egg-rr 53.2%

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

   5. Taylor expanded in x around inf 75.5%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

      3. unpow2 75.6%

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

      4. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}\\ t\_0 \cdot \left(t\_0 \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot x (sqrt x)))))
   (* t_0 (* t_0 (/ 1.0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))))

C

double code(double x) {
	double t_0 = 1.0 / hypot(x, sqrt(x));
	return t_0 * (t_0 * (1.0 / (pow(x, -0.5) + pow((1.0 + x), -0.5))));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.hypot(x, Math.sqrt(x));
	return t_0 * (t_0 * (1.0 / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5))));
}

Python

def code(x):
	t_0 = 1.0 / math.hypot(x, math.sqrt(x))
	return t_0 * (t_0 * (1.0 / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))))

Julia

function code(x)
	t_0 = Float64(1.0 / hypot(x, sqrt(x)))
	return Float64(t_0 * Float64(t_0 * Float64(1.0 / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / hypot(x, sqrt(x));
	tmp = t_0 * (t_0 * (1.0 / ((x ^ -0.5) + ((1.0 + x) ^ -0.5))));
end

Wolfram

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

Derivation

1. Initial program 36.4%

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

   1. flip-- 36.4%

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

   2. div-inv 36.4%

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

   3. frac-times 22.8%

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

   4. metadata-eval 22.8%

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

   5. add-sqr-sqrt 19.7%

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

   6. frac-times 26.1%

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

   7. metadata-eval 26.1%

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

   8. add-sqr-sqrt 36.5%

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

   9. +-commutative 36.5%

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

   10. inv-pow 36.5%

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

   11. sqrt-pow2 36.5%

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

   12. metadata-eval 36.5%

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

   13. pow1/2 36.5%

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

   14. pow-flip 36.5%

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

   15. +-commutative 36.5%

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

   16. metadata-eval 36.5%

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

4. Applied egg-rr 36.5%

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

   1. *-commutative 36.5%

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

   2. frac-sub 38.3%

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

   3. frac-times 38.3%

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

   4. *-un-lft-identity 38.3%

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

   5. *-un-lft-identity 38.3%

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

   6. *-rgt-identity 38.3%

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

   7. distribute-lft-in 38.3%

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

   8. *-rgt-identity 38.3%

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

   9. pow2 38.3%

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

6. Applied egg-rr 38.3%

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

   1. *-commutative 38.3%

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

   2. associate-/r* 38.3%

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

   3. associate--l+ 85.4%

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

   4. +-inverses 85.4%

      \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   5. metadata-eval 85.4%

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

   6. +-commutative 85.4%

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

   7. unpow2 85.4%

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

   8. fma-define 85.4%

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

8. Simplified 85.4%

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

   1. div-inv 85.3%

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

   2. add-sqr-sqrt 85.2%

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

   3. associate-*l* 85.1%

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

   4. sqrt-div 85.2%

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

   5. metadata-eval 85.2%

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

   6. fma-undefine 85.2%

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

   7. add-sqr-sqrt 85.2%

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

   8. hypot-define 85.2%

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

   9. sqrt-div 85.1%

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

   10. metadata-eval 85.1%

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

   11. fma-undefine 85.1%

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

   12. add-sqr-sqrt 85.1%

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

   13. hypot-define 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+113)
   (/ (/ (/ 1.0 x) (+ 1.0 x)) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))
   (* (/ 1.0 (hypot x (sqrt x))) (* 0.5 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+113) {
		tmp = ((1.0 / x) / (1.0 + x)) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 / hypot(x, sqrt(x))) * (0.5 * sqrt((1.0 / x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+113) {
		tmp = ((1.0 / x) / (1.0 + x)) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 / Math.hypot(x, Math.sqrt(x))) * (0.5 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e+113:
		tmp = ((1.0 / x) / (1.0 + x)) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	else:
		tmp = (1.0 / math.hypot(x, math.sqrt(x))) * (0.5 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+113)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(1.0 + x)) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / hypot(x, sqrt(x))) * Float64(0.5 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e+113)
		tmp = ((1.0 / x) / (1.0 + x)) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	else
		tmp = (1.0 / hypot(x, sqrt(x))) * (0.5 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e+113], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e113

   1. Initial program 12.5%

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

      1. flip-- 12.5%

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

      2. div-inv 12.5%

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

      3. frac-times 12.7%

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

      4. metadata-eval 12.7%

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

      5. add-sqr-sqrt 12.9%

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

      6. frac-times 13.1%

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

      7. metadata-eval 13.1%

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

      8. add-sqr-sqrt 12.9%

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

      9. +-commutative 12.9%

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

      10. inv-pow 12.9%

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

      11. sqrt-pow2 12.9%

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

      12. metadata-eval 12.9%

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

      13. pow1/2 12.9%

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

      14. pow-flip 12.9%

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

      15. +-commutative 12.9%

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

      16. metadata-eval 12.9%

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

   4. Applied egg-rr 12.9%

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

      1. *-commutative 12.9%

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

      2. frac-sub 17.4%

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

      3. frac-times 17.4%

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

      4. *-un-lft-identity 17.4%

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

      5. *-un-lft-identity 17.4%

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

      6. *-rgt-identity 17.4%

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

      7. distribute-lft-in 17.4%

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

      8. *-rgt-identity 17.4%

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

      9. pow2 17.4%

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

   6. Applied egg-rr 17.4%

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

      1. *-commutative 17.4%

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

      2. associate-/r* 17.4%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

      7. unpow2 99.6%

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

      8. fma-define 99.6%

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

   8. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. fma-undefine 99.6%

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

      3. *-rgt-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 99.3%

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

      7. inv-pow 99.3%

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

      8. metadata-eval 99.3%

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

      9. pow-prod-up 99.3%

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

      10. associate-*l* 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. associate-*r* 99.3%

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

       2. pow-sqr 99.3%

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

       3. metadata-eval 99.3%

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

       4. unpow-1 99.3%

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

       5. associate-*r/ 99.5%

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

       6. *-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

   if 2e113 < x

   1. Initial program 51.7%

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

      1. flip-- 51.7%

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

      2. div-inv 51.7%

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

      3. frac-times 29.2%

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

      4. metadata-eval 29.2%

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

      5. add-sqr-sqrt 24.0%

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

      6. frac-times 34.5%

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

      7. metadata-eval 34.5%

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

      8. add-sqr-sqrt 51.7%

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

      9. +-commutative 51.7%

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

      10. inv-pow 51.7%

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

      11. sqrt-pow2 51.7%

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

      12. metadata-eval 51.7%

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

      13. pow1/2 51.7%

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

      14. pow-flip 51.7%

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

      15. +-commutative 51.7%

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

      16. metadata-eval 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. *-commutative 51.7%

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

      2. frac-sub 51.7%

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

      3. frac-times 51.7%

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

      4. *-un-lft-identity 51.7%

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

      5. *-un-lft-identity 51.7%

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

      6. *-rgt-identity 51.7%

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

      7. distribute-lft-in 51.7%

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

      8. *-rgt-identity 51.7%

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

      9. pow2 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-/r* 51.7%

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

      3. associate--l+ 76.3%

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

      4. +-inverses 76.3%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      5. metadata-eval 76.3%

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

      6. +-commutative 76.3%

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

      7. unpow2 76.3%

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

      8. fma-define 76.3%

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

   8. Simplified 76.3%

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

      1. div-inv 76.3%

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

      2. add-sqr-sqrt 76.2%

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

      3. associate-*l* 76.2%

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

      4. sqrt-div 76.2%

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

      5. metadata-eval 76.2%

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

      6. fma-undefine 76.2%

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

      7. add-sqr-sqrt 76.2%

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

      8. hypot-define 76.2%

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

      9. sqrt-div 76.2%

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

      10. metadata-eval 76.2%

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

      11. fma-undefine 76.2%

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

      12. add-sqr-sqrt 76.2%

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

      13. hypot-define 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* x (fma (pow x -0.5) (+ 1.0 x) (pow (+ 1.0 x) 0.5)))))

C

double code(double x) {
	return 1.0 / (x * fma(pow(x, -0.5), (1.0 + x), pow((1.0 + x), 0.5)));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * fma((x ^ -0.5), Float64(1.0 + x), (Float64(1.0 + x) ^ 0.5))))
end

Wolfram

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

Derivation

1. Initial program 36.4%

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

   1. flip-- 36.4%

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

   2. div-inv 36.4%

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

   3. frac-times 22.8%

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

   4. metadata-eval 22.8%

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

   5. add-sqr-sqrt 19.7%

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

   6. frac-times 26.1%

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

   7. metadata-eval 26.1%

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

   8. add-sqr-sqrt 36.5%

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

   9. +-commutative 36.5%

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

   10. inv-pow 36.5%

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

   11. sqrt-pow2 36.5%

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

   12. metadata-eval 36.5%

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

   13. pow1/2 36.5%

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

   14. pow-flip 36.5%

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

   15. +-commutative 36.5%

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

   16. metadata-eval 36.5%

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

4. Applied egg-rr 36.5%

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

   1. *-commutative 36.5%

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

   2. frac-sub 38.3%

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

   3. frac-times 38.3%

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

   4. *-un-lft-identity 38.3%

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

   5. *-un-lft-identity 38.3%

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

   6. *-rgt-identity 38.3%

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

   7. distribute-lft-in 38.3%

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

   8. *-rgt-identity 38.3%

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

   9. pow2 38.3%

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

6. Applied egg-rr 38.3%

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

   1. *-commutative 38.3%

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

   2. associate-/r* 38.3%

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

   3. associate--l+ 85.4%

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

   4. +-inverses 85.4%

      \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   5. metadata-eval 85.4%

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

   6. +-commutative 85.4%

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

   7. unpow2 85.4%

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

   8. fma-define 85.4%

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

8. Simplified 85.4%

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

   1. div-inv 85.3%

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

   2. add-sqr-sqrt 85.2%

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

   3. associate-*l* 85.1%

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

   4. sqrt-div 85.2%

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

   5. metadata-eval 85.2%

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

   6. fma-undefine 85.2%

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

   7. add-sqr-sqrt 85.2%

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

   8. hypot-define 85.2%

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

   9. sqrt-div 85.1%

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

   10. metadata-eval 85.1%

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

   11. fma-undefine 85.1%

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

   12. add-sqr-sqrt 85.1%

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

   13. hypot-define 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. frac-times 99.5%

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

    3. metadata-eval 99.5%

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

    4. frac-times 98.7%

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

    5. metadata-eval 98.7%

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

12. Applied egg-rr 98.7%

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

    1. *-commutative 98.7%

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

    2. associate-*r* 85.3%

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

    3. hypot-undefine 85.3%

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

    4. hypot-undefine 85.3%

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

    5. rem-square-sqrt 85.3%

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

    6. rem-square-sqrt 85.3%

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

    7. fma-undefine 85.3%

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

    8. fma-undefine 85.3%

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

    9. *-rgt-identity 85.3%

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

    10. distribute-lft-in 85.3%

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

    11. +-commutative 85.3%

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

    12. associate-*l* 98.7%

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

    13. distribute-rgt-in 98.7%

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

    14. fma-define 98.7%

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

    15. pow-plus 98.7%

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

    16. metadata-eval 98.7%

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

14. Simplified 98.7%

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

15. Final simplification 98.7%

    \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)} \]
16. Add Preprocessing

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

   1. flip-- 36.4%

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

   2. div-inv 36.4%

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

   3. frac-times 22.8%

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

   4. metadata-eval 22.8%

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

   5. add-sqr-sqrt 19.7%

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

   6. frac-times 26.1%

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

   7. metadata-eval 26.1%

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

   8. add-sqr-sqrt 36.5%

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

   9. +-commutative 36.5%

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

   10. inv-pow 36.5%

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

   11. sqrt-pow2 36.5%

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

   12. metadata-eval 36.5%

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

   13. pow1/2 36.5%

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

   14. pow-flip 36.5%

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

   15. +-commutative 36.5%

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

   16. metadata-eval 36.5%

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

4. Applied egg-rr 36.5%

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

5. Taylor expanded in x around inf 82.9%

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

   1. associate-*l/ 82.9%

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

   2. *-un-lft-identity 82.9%

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

   3. unpow2 82.9%

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

   4. associate-/r* 97.1%

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

7. Applied egg-rr 97.1%

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

8. Final simplification 97.1%

   \[\leadsto \frac{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{x}}{x} \]
9. Add Preprocessing

Alternative 6: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{x}}{1 + x}}{\sqrt{\frac{1}{x}} \cdot 2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ 1.0 x) (+ 1.0 x)) (* (sqrt (/ 1.0 x)) 2.0)))

C

double code(double x) {
	return ((1.0 / x) / (1.0 + x)) / (sqrt((1.0 / x)) * 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return ((1.0 / x) / (1.0 + x)) / (Math.sqrt((1.0 / x)) * 2.0);
}

Python

def code(x):
	return ((1.0 / x) / (1.0 + x)) / (math.sqrt((1.0 / x)) * 2.0)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) / Float64(1.0 + x)) / Float64(sqrt(Float64(1.0 / x)) * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / x) / (1.0 + x)) / (sqrt((1.0 / x)) * 2.0);
end

Wolfram

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

Derivation

1. Initial program 36.4%

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

   1. flip-- 36.4%

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

   2. div-inv 36.4%

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

   3. frac-times 22.8%

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

   4. metadata-eval 22.8%

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

   5. add-sqr-sqrt 19.7%

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

   6. frac-times 26.1%

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

   7. metadata-eval 26.1%

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

   8. add-sqr-sqrt 36.5%

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

   9. +-commutative 36.5%

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

   10. inv-pow 36.5%

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

   11. sqrt-pow2 36.5%

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

   12. metadata-eval 36.5%

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

   13. pow1/2 36.5%

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

   14. pow-flip 36.5%

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

   15. +-commutative 36.5%

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

   16. metadata-eval 36.5%

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

4. Applied egg-rr 36.5%

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

   1. *-commutative 36.5%

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

   2. frac-sub 38.3%

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

   3. frac-times 38.3%

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

   4. *-un-lft-identity 38.3%

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

   5. *-un-lft-identity 38.3%

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

   6. *-rgt-identity 38.3%

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

   7. distribute-lft-in 38.3%

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

   8. *-rgt-identity 38.3%

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

   9. pow2 38.3%

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

6. Applied egg-rr 38.3%

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

   1. *-commutative 38.3%

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

   2. associate-/r* 38.3%

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

   3. associate--l+ 85.4%

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

   4. +-inverses 85.4%

      \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   5. metadata-eval 85.4%

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

   6. +-commutative 85.4%

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

   7. unpow2 85.4%

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

   8. fma-define 85.4%

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

8. Simplified 85.4%

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

9. Taylor expanded in x around inf 83.1%

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

    1. metadata-eval 85.4%

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

    2. fma-undefine 85.4%

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

    3. *-rgt-identity 85.4%

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

    4. distribute-lft-in 85.4%

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

    5. +-commutative 85.4%

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

    6. frac-times 85.9%

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

    7. inv-pow 85.9%

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

    8. metadata-eval 85.9%

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

    9. pow-prod-up 85.9%

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

    10. associate-*l* 85.9%

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

11. Applied egg-rr 83.6%

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

    1. associate-*r* 85.9%

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

    2. pow-sqr 85.9%

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

    3. metadata-eval 85.9%

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

    4. unpow-1 85.9%

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

    5. associate-*r/ 86.0%

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

    6. *-rgt-identity 86.0%

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

13. Simplified 83.7%

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

14. Final simplification 83.7%

    \[\leadsto \frac{\frac{\frac{1}{x}}{1 + x}}{\sqrt{\frac{1}{x}} \cdot 2} \]
15. Add Preprocessing

Alternative 7: 37.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (/ 1.0 (+ x (pow x 0.5))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + pow(x, 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / (x + (x ** 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + Math.pow(x, 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / (x + math.pow(x, 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / Float64(x + (x ^ 0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / (x + (x ^ 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / N[(x + N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 10.5%

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

      1. flip-- 10.5%

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

      2. div-inv 10.5%

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

      3. frac-times 10.4%

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

      4. metadata-eval 10.4%

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

      5. add-sqr-sqrt 10.5%

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

      6. frac-times 10.8%

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

      7. metadata-eval 10.8%

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

      8. add-sqr-sqrt 10.8%

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

      9. +-commutative 10.8%

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

      10. inv-pow 10.8%

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

      11. sqrt-pow2 10.8%

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

      12. metadata-eval 10.8%

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

      13. pow1/2 10.8%

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

      14. pow-flip 10.8%

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

      15. +-commutative 10.8%

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

      16. metadata-eval 10.8%

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

   4. Applied egg-rr 10.8%

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

   5. Taylor expanded in x around 0 8.4%

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

      1. distribute-rgt-in 8.4%

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

      2. *-lft-identity 8.4%

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

      3. pow-plus 8.4%

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

      4. metadata-eval 8.4%

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

   7. Simplified 8.4%

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

   if 4.6000000000000003e153 < x

   1. Initial program 67.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 - x\right)} \cdot \left(\sqrt{x} \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ -1.0 (* x (- -1.0 x))) (* (sqrt x) 0.5)))

C

double code(double x) {
	return (-1.0 / (x * (-1.0 - x))) * (sqrt(x) * 0.5);
}

Fortran

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

Java

public static double code(double x) {
	return (-1.0 / (x * (-1.0 - x))) * (Math.sqrt(x) * 0.5);
}

Python

def code(x):
	return (-1.0 / (x * (-1.0 - x))) * (math.sqrt(x) * 0.5)

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(x * Float64(-1.0 - x))) * Float64(sqrt(x) * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / (x * (-1.0 - x))) * (sqrt(x) * 0.5);
end

Wolfram

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

Derivation

1. Initial program 36.4%

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

   1. flip-- 36.4%

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

   2. div-inv 36.4%

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

   3. frac-times 22.8%

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

   4. metadata-eval 22.8%

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

   5. add-sqr-sqrt 19.7%

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

   6. frac-times 26.1%

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

   7. metadata-eval 26.1%

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

   8. add-sqr-sqrt 36.5%

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

   9. +-commutative 36.5%

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

   10. inv-pow 36.5%

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

   11. sqrt-pow2 36.5%

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

   12. metadata-eval 36.5%

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

   13. pow1/2 36.5%

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

   14. pow-flip 36.5%

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

   15. +-commutative 36.5%

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

   16. metadata-eval 36.5%

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

4. Applied egg-rr 36.5%

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

5. Taylor expanded in x around inf 35.2%

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

   1. frac-2neg 35.2%

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

   2. metadata-eval 35.2%

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

   3. frac-sub 36.0%

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

   4. *-un-lft-identity 36.0%

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

7. Applied egg-rr 36.0%

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

8. Taylor expanded in x around 0 83.1%

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

9. Final simplification 83.1%

   \[\leadsto \frac{-1}{x \cdot \left(-1 - x\right)} \cdot \left(\sqrt{x} \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 9: 35.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 36.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 33.0%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 33.0%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 98.2% 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 2024052 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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