2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.9%

Time: 12.2s

Alternatives: 5

Speedup: 2.0×

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/ (/ (/ -1.0 x) (- -1.0 x)) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((-1.0 / x) / (-1.0 - x)) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (((-1.0d0) / x) / ((-1.0d0) - x)) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((-1.0 / x) / (-1.0 - x)) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = ((-1.0 / x) / (-1.0 - x)) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 36.9%

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

      1. flip-- 36.9%

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

      2. clear-num 36.9%

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

      3. pow1/2 36.9%

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

      4. pow-flip 36.9%

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

      5. metadata-eval 36.9%

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

      6. inv-pow 36.9%

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

      7. sqrt-pow2 36.9%

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

      8. +-commutative 36.9%

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

      9. metadata-eval 36.9%

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

      10. frac-times 19.3%

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

      11. metadata-eval 19.3%

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

      12. add-sqr-sqrt 18.4%

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

      13. frac-times 23.2%

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

      14. metadata-eval 23.2%

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

      15. add-sqr-sqrt 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. frac-2neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. frac-sub 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. *-lft-identity 36.9%

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

      2. neg-mul-1 36.9%

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

      3. *-commutative 36.9%

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

      4. /-rgt-identity 36.9%

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

      5. neg-mul-1 36.9%

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

      6. *-rgt-identity 36.9%

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

      7. distribute-rgt-neg-in 36.9%

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

      8. distribute-lft-neg-in 36.9%

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

      9. /-rgt-identity 36.9%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in x around inf 66.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   10. Step-by-step derivation

       1. unpow-1 66.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

       2. exp-to-pow 63.6%

          \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

       3. *-commutative 63.6%

          \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

       4. exp-prod 64.8%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

       5. *-commutative 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

       6. associate-*r* 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

       7. metadata-eval 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

       8. *-commutative 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

       9. exp-to-pow 67.3%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

       10. metadata-eval 67.3%

           \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

       11. pow-sqr 67.4%

           \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

       12. rem-sqrt-square 100.0%

           \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

       13. rem-square-sqrt 99.5%

           \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

       14. fabs-sqr 99.5%

           \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

       15. rem-square-sqrt 100.0%

           \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 57.2%

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

      1. flip-- 57.7%

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

      2. clear-num 57.7%

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

      3. pow1/2 57.7%

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

      4. pow-flip 57.7%

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

      5. metadata-eval 57.7%

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

      6. inv-pow 57.7%

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

      7. sqrt-pow2 57.7%

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

      8. +-commutative 57.7%

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

      9. metadata-eval 57.7%

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

      10. frac-times 57.4%

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

      11. metadata-eval 57.4%

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

      12. add-sqr-sqrt 58.1%

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

      13. frac-times 58.1%

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

      14. metadata-eval 58.1%

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

      15. add-sqr-sqrt 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. frac-2neg 59.1%

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

      2. metadata-eval 59.1%

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

      3. frac-sub 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

      2. neg-mul-1 99.3%

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

      3. *-commutative 99.3%

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

      4. /-rgt-identity 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. distribute-lft-neg-in 99.3%

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

      9. /-rgt-identity 99.3%

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

   8. Simplified 99.3%

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

      1. clear-num 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. times-frac 99.4%

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

      5. metadata-eval 99.4%

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

      6. +-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 1.7%

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

       3. frac-times 1.7%

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

       4. metadata-eval 1.7%

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

       5. metadata-eval 1.7%

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

       6. frac-times 1.7%

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

       7. sqrt-unprod 1.7%

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

       8. add-sqr-sqrt 1.7%

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

       9. expm1-log1p-u 1.7%

          \[\leadsto -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{x \cdot \left(-1 - x\right)}\right)\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]

       10. expm1-udef 2.5%

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

   12. Applied egg-rr 35.5%

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

       1. expm1-def 99.5%

          \[\leadsto -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{x}}{-1 - x}\right)\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]

       2. expm1-log1p 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x \cdot \left(-1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/ -1.0 (* (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)) (* x (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = -1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) * (x * (-1.0 - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (-1.0d0) / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) * (x * ((-1.0d0) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = -1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) * (x * (-1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = -1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) * (x * (-1.0 - x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 36.9%

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

      1. flip-- 36.9%

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

      2. clear-num 36.9%

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

      3. pow1/2 36.9%

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

      4. pow-flip 36.9%

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

      5. metadata-eval 36.9%

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

      6. inv-pow 36.9%

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

      7. sqrt-pow2 36.9%

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

      8. +-commutative 36.9%

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

      9. metadata-eval 36.9%

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

      10. frac-times 19.3%

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

      11. metadata-eval 19.3%

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

      12. add-sqr-sqrt 18.4%

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

      13. frac-times 23.2%

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

      14. metadata-eval 23.2%

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

      15. add-sqr-sqrt 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. frac-2neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. frac-sub 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. *-lft-identity 36.9%

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

      2. neg-mul-1 36.9%

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

      3. *-commutative 36.9%

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

      4. /-rgt-identity 36.9%

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

      5. neg-mul-1 36.9%

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

      6. *-rgt-identity 36.9%

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

      7. distribute-rgt-neg-in 36.9%

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

      8. distribute-lft-neg-in 36.9%

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

      9. /-rgt-identity 36.9%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in x around inf 66.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   10. Step-by-step derivation

       1. unpow-1 66.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

       2. exp-to-pow 63.6%

          \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

       3. *-commutative 63.6%

          \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

       4. exp-prod 64.8%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

       5. *-commutative 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

       6. associate-*r* 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

       7. metadata-eval 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

       8. *-commutative 64.8%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

       9. exp-to-pow 67.3%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

       10. metadata-eval 67.3%

           \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

       11. pow-sqr 67.4%

           \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

       12. rem-sqrt-square 100.0%

           \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

       13. rem-square-sqrt 99.5%

           \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

       14. fabs-sqr 99.5%

           \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

       15. rem-square-sqrt 100.0%

           \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 57.2%

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

      1. flip-- 57.7%

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

      2. clear-num 57.7%

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

      3. pow1/2 57.7%

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

      4. pow-flip 57.7%

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

      5. metadata-eval 57.7%

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

      6. inv-pow 57.7%

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

      7. sqrt-pow2 57.7%

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

      8. +-commutative 57.7%

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

      9. metadata-eval 57.7%

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

      10. frac-times 57.4%

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

      11. metadata-eval 57.4%

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

      12. add-sqr-sqrt 58.1%

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

      13. frac-times 58.1%

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

      14. metadata-eval 58.1%

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

      15. add-sqr-sqrt 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. frac-2neg 59.1%

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

      2. metadata-eval 59.1%

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

      3. frac-sub 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

      2. neg-mul-1 99.3%

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

      3. *-commutative 99.3%

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

      4. /-rgt-identity 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. distribute-lft-neg-in 99.3%

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

      9. /-rgt-identity 99.3%

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

   8. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{-1}{x \cdot \left(-1 - x\right)}}}\right)\right)} \]

      2. expm1-udef 42.4%

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

      3. clear-num 42.4%

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

      4. associate-/l/ 42.4%

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

      5. +-commutative 42.4%

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

   10. Applied egg-rr 42.4%

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

       1. expm1-def 99.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x \cdot \left(-1 - x\right)\right)}\right)\right)} \]

       2. expm1-log1p 99.4%

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

   12. Simplified 99.4%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x \cdot \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 3.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

   1. add-exp-log 7.2%

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

   2. log-rec 7.2%

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

   3. pow1/2 7.2%

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

   4. log-pow 7.2%

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

   5. +-commutative 7.2%

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

   6. log1p-udef 7.2%

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

4. Applied egg-rr 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. metadata-eval 7.2%

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

   3. exp-prod 6.8%

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

6. Simplified 6.8%

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

7. Taylor expanded in x around inf 3.2%

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

   1. mul-1-neg 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto -\sqrt{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 4: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.9%

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

   1. flip-- 37.9%

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

   2. clear-num 38.0%

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

   3. pow1/2 38.0%

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

   4. pow-flip 38.0%

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

   5. metadata-eval 38.0%

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

   6. inv-pow 38.0%

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

   7. sqrt-pow2 38.0%

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

   8. +-commutative 38.0%

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

   9. metadata-eval 38.0%

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

   10. frac-times 21.2%

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

   11. metadata-eval 21.2%

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

   12. add-sqr-sqrt 20.4%

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

   13. frac-times 24.9%

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

   14. metadata-eval 24.9%

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

   15. add-sqr-sqrt 38.0%

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

4. Applied egg-rr 38.0%

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

   1. frac-2neg 38.0%

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

   2. metadata-eval 38.0%

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

   3. frac-sub 40.1%

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

6. Applied egg-rr 40.1%

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

   1. *-lft-identity 40.1%

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

   2. neg-mul-1 40.1%

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

   3. *-commutative 40.1%

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

   4. /-rgt-identity 40.1%

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

   5. neg-mul-1 40.1%

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

   6. *-rgt-identity 40.1%

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

   7. distribute-rgt-neg-in 40.1%

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

   8. distribute-lft-neg-in 40.1%

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

   9. /-rgt-identity 40.1%

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

8. Simplified 84.5%

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

9. Taylor expanded in x around inf 65.8%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
10. Step-by-step derivation

    1. unpow-1 65.8%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

    2. exp-to-pow 63.5%

       \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

    3. *-commutative 63.5%

       \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

    4. exp-prod 64.6%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

    5. *-commutative 64.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

    6. associate-*r* 64.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

    7. metadata-eval 64.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

    8. *-commutative 64.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

    9. exp-to-pow 67.0%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

    10. metadata-eval 67.0%

        \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

    11. pow-sqr 67.1%

        \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

    12. rem-sqrt-square 98.0%

        \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

    13. rem-square-sqrt 97.5%

        \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

    14. fabs-sqr 97.5%

        \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

    15. rem-square-sqrt 98.0%

        \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

11. Simplified 98.0%

    \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

12. Final simplification 98.0%

    \[\leadsto 0.5 \cdot {x}^{-1.5} \]
13. Add Preprocessing

Alternative 5: 2.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around 0 2.5%

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

4. Taylor expanded in x around inf 2.5%

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

5. Final simplification 2.5%

   \[\leadsto -1 \]
6. Add Preprocessing

Developer target: 98.5% 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 2024026 
(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)))))