2isqrt (example 3.6)

Percentage Accurate: 69.4% → 99.8%

Time: 9.8s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 1e-6)
   (/
    (-
     (+ (/ 0.5 x) (/ 0.3125 (pow x 3.0)))
     (+ (/ 0.375 (* x x)) (/ 0.2734375 (pow x 4.0))))
    (sqrt x))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-6) {
		tmp = (((0.5 / x) + (0.3125 / pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / pow(x, 4.0)))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 1d-6) then
        tmp = (((0.5d0 / x) + (0.3125d0 / (x ** 3.0d0))) - ((0.375d0 / (x * x)) + (0.2734375d0 / (x ** 4.0d0)))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 1e-6) {
		tmp = (((0.5 / x) + (0.3125 / Math.pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / Math.pow(x, 4.0)))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 1e-6:
		tmp = (((0.5 / x) + (0.3125 / math.pow(x, 3.0))) - ((0.375 / (x * x)) + (0.2734375 / math.pow(x, 4.0)))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 1e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 / x) + Float64(0.3125 / (x ^ 3.0))) - Float64(Float64(0.375 / Float64(x * x)) + Float64(0.2734375 / (x ^ 4.0)))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-6)
		tmp = (((0.5 / x) + (0.3125 / (x ^ 3.0))) - ((0.375 / (x * x)) + (0.2734375 / (x ^ 4.0)))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(N[(0.5 / x), $MachinePrecision] + N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.2734375 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.99999999999999955e-7

   1. Initial program 33.1%

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

      1. frac-sub 33.1%

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

      2. div-inv 33.1%

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

      3. *-un-lft-identity 33.1%

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

      4. +-commutative 33.1%

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

      5. *-rgt-identity 33.1%

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

      6. metadata-eval 33.1%

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

      7. frac-times 33.1%

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

      8. un-div-inv 33.1%

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

      9. pow1/2 33.1%

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

      10. pow-flip 33.1%

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

      11. metadata-eval 33.1%

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

      12. +-commutative 33.1%

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

   3. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

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

      2. remove-double-neg 33.1%

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

      3. neg-mul-1 33.1%

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

      4. *-commutative 33.1%

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

      5. times-frac 33.1%

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

   5. Simplified 33.1%

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

      1. associate-*r/ 33.1%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. sub-neg 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.8%

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

      8. sqr-neg 5.8%

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

      9. sqrt-prod 5.8%

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

      10. add-sqr-sqrt 5.8%

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

      11. remove-double-neg 5.8%

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

      12. frac-2neg 5.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 33.1%

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

   7. Applied egg-rr 33.3%

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

      1. associate-/r/ 33.3%

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

      2. associate-*l/ 33.3%

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

      3. neg-mul-1 33.3%

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

      4. distribute-rgt-neg-in 33.3%

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

      5. *-lft-identity 33.3%

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

      6. distribute-neg-in 33.3%

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

      7. metadata-eval 33.3%

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

      8. unsub-neg 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. associate-*r/ 99.6%

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

       2. metadata-eval 99.6%

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

       3. associate-*r/ 99.6%

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

       4. metadata-eval 99.6%

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

       5. +-commutative 99.6%

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

       6. associate-*r/ 99.6%

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

       7. metadata-eval 99.6%

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

       8. unpow2 99.6%

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

       9. associate-*r/ 99.6%

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

       10. metadata-eval 99.6%

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

   12. Simplified 99.6%

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

   if 9.99999999999999955e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 1e-6)
   (/ (+ (/ 0.5 x) (- (/ 0.3125 (pow x 3.0)) (/ 0.375 (* x x)))) (sqrt x))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-6) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) - (0.375 / (x * x)))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 1d-6) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) - (0.375d0 / (x * x)))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 1e-6) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) - (0.375 / (x * x)))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 1e-6:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) - (0.375 / (x * x)))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 1e-6)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) - Float64(0.375 / Float64(x * x)))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-6)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) - (0.375 / (x * x)))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.99999999999999955e-7

   1. Initial program 33.1%

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

      1. frac-sub 33.1%

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

      2. div-inv 33.1%

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

      3. *-un-lft-identity 33.1%

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

      4. +-commutative 33.1%

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

      5. *-rgt-identity 33.1%

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

      6. metadata-eval 33.1%

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

      7. frac-times 33.1%

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

      8. un-div-inv 33.1%

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

      9. pow1/2 33.1%

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

      10. pow-flip 33.1%

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

      11. metadata-eval 33.1%

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

      12. +-commutative 33.1%

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

   3. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

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

      2. remove-double-neg 33.1%

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

      3. neg-mul-1 33.1%

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

      4. *-commutative 33.1%

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

      5. times-frac 33.1%

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

   5. Simplified 33.1%

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

      1. associate-*r/ 33.1%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. sub-neg 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.8%

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

      8. sqr-neg 5.8%

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

      9. sqrt-prod 5.8%

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

      10. add-sqr-sqrt 5.8%

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

      11. remove-double-neg 5.8%

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

      12. frac-2neg 5.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 33.1%

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

   7. Applied egg-rr 33.3%

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

      1. associate-/r/ 33.3%

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

      2. associate-*l/ 33.3%

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

      3. neg-mul-1 33.3%

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

      4. distribute-rgt-neg-in 33.3%

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

      5. *-lft-identity 33.3%

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

      6. distribute-neg-in 33.3%

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

      7. metadata-eval 33.3%

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

      8. unsub-neg 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate--l+ 99.5%

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

       2. associate-*r/ 99.5%

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

       3. metadata-eval 99.5%

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

       4. associate-*r/ 99.5%

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

       5. metadata-eval 99.5%

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

       6. associate-*r/ 99.5%

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

       7. metadata-eval 99.5%

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

       8. unpow2 99.5%

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

   12. Simplified 99.5%

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

   if 9.99999999999999955e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (/ (- (/ 0.5 x) (/ 0.375 (* x x))) (sqrt x))
   (- (pow x -0.5) (/ (/ 1.0 (+ 1.0 x)) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 2e-9) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - ((1.0 / (1.0 + x)) / pow((1.0 + x), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 2d-9) then
        tmp = ((0.5d0 / x) - (0.375d0 / (x * x))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 / (1.0d0 + x)) / ((1.0d0 + x) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 2e-9) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - ((1.0 / (1.0 + x)) / Math.pow((1.0 + x), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 2e-9:
		tmp = ((0.5 / x) - (0.375 / (x * x))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - ((1.0 / (1.0 + x)) / math.pow((1.0 + x), -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 2e-9)
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - Float64(Float64(1.0 / Float64(1.0 + x)) / (Float64(1.0 + x) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 2e-9)
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((1.0 / (1.0 + x)) / ((1.0 + x) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-9], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $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)))) < 2.00000000000000012e-9

   1. Initial program 32.7%

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

      1. frac-sub 32.7%

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

      2. div-inv 32.7%

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

      3. *-un-lft-identity 32.7%

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

      4. +-commutative 32.7%

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

      5. *-rgt-identity 32.7%

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

      6. metadata-eval 32.7%

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

      7. frac-times 32.7%

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

      8. un-div-inv 32.7%

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

      9. pow1/2 32.7%

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

      10. pow-flip 32.7%

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

      11. metadata-eval 32.7%

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

      12. +-commutative 32.7%

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

   3. Applied egg-rr 32.7%

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

      1. associate-*r/ 32.7%

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

      2. remove-double-neg 32.7%

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

      3. neg-mul-1 32.7%

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

      4. *-commutative 32.7%

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

      5. times-frac 32.7%

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

   5. Simplified 32.7%

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

      1. associate-*r/ 32.7%

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

      2. clear-num 32.7%

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

      3. *-commutative 32.7%

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

      4. sub-neg 32.7%

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

      5. distribute-neg-frac 32.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.7%

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

      8. sqr-neg 5.7%

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

      9. sqrt-prod 5.7%

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

      10. add-sqr-sqrt 5.7%

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

      11. remove-double-neg 5.7%

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

      12. frac-2neg 5.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 32.7%

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

   7. Applied egg-rr 32.8%

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

      1. associate-/r/ 32.8%

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

      2. associate-*l/ 32.8%

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

      3. neg-mul-1 32.8%

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

      4. distribute-rgt-neg-in 32.8%

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

      5. *-lft-identity 32.8%

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

      6. distribute-neg-in 32.8%

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

      7. metadata-eval 32.8%

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

      8. unsub-neg 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in x around inf 99.4%

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

       1. associate-*r/ 99.4%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       2. metadata-eval 99.4%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       3. associate-*r/ 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}}{\sqrt{x}} \]

       4. metadata-eval 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}}{\sqrt{x}} \]

       5. unpow2 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}}{\sqrt{x}} \]

   12. Simplified 99.4%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}}{\sqrt{x}} \]

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. distribute-lft1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. mul0-lft 99.8%

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

      5. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

      1. pow-to-exp 99.8%

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

      2. log1p-udef 99.8%

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

   7. Applied egg-rr 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. sub0-neg 99.8%

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

       2. distribute-neg-frac 99.8%

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

       3. metadata-eval 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.6%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (/ (- (/ 0.5 x) (/ 0.375 (* x x))) (sqrt x))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 2e-9) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 2d-9) then
        tmp = ((0.5d0 / x) - (0.375d0 / (x * x))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 2e-9) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 2e-9:
		tmp = ((0.5 / x) - (0.375 / (x * x))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 2e-9)
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 2e-9)
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-9], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.00000000000000012e-9

   1. Initial program 32.7%

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

      1. frac-sub 32.7%

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

      2. div-inv 32.7%

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

      3. *-un-lft-identity 32.7%

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

      4. +-commutative 32.7%

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

      5. *-rgt-identity 32.7%

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

      6. metadata-eval 32.7%

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

      7. frac-times 32.7%

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

      8. un-div-inv 32.7%

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

      9. pow1/2 32.7%

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

      10. pow-flip 32.7%

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

      11. metadata-eval 32.7%

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

      12. +-commutative 32.7%

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

   3. Applied egg-rr 32.7%

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

      1. associate-*r/ 32.7%

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

      2. remove-double-neg 32.7%

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

      3. neg-mul-1 32.7%

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

      4. *-commutative 32.7%

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

      5. times-frac 32.7%

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

   5. Simplified 32.7%

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

      1. associate-*r/ 32.7%

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

      2. clear-num 32.7%

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

      3. *-commutative 32.7%

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

      4. sub-neg 32.7%

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

      5. distribute-neg-frac 32.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.7%

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

      8. sqr-neg 5.7%

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

      9. sqrt-prod 5.7%

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

      10. add-sqr-sqrt 5.7%

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

      11. remove-double-neg 5.7%

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

      12. frac-2neg 5.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 32.7%

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

   7. Applied egg-rr 32.8%

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

      1. associate-/r/ 32.8%

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

      2. associate-*l/ 32.8%

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

      3. neg-mul-1 32.8%

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

      4. distribute-rgt-neg-in 32.8%

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

      5. *-lft-identity 32.8%

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

      6. distribute-neg-in 32.8%

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

      7. metadata-eval 32.8%

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

      8. unsub-neg 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in x around inf 99.4%

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

       1. associate-*r/ 99.4%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       2. metadata-eval 99.4%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       3. associate-*r/ 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}}{\sqrt{x}} \]

       4. metadata-eval 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}}{\sqrt{x}} \]

       5. unpow2 99.4%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}}{\sqrt{x}} \]

   12. Simplified 99.4%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}}{\sqrt{x}} \]

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. distribute-lft1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. mul0-lft 99.8%

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

      5. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.2%

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

   if 1.1000000000000001 < x

   1. Initial program 33.1%

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

      1. frac-sub 33.1%

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

      2. div-inv 33.1%

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

      3. *-un-lft-identity 33.1%

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

      4. +-commutative 33.1%

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

      5. *-rgt-identity 33.1%

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

      6. metadata-eval 33.1%

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

      7. frac-times 33.1%

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

      8. un-div-inv 33.1%

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

      9. pow1/2 33.1%

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

      10. pow-flip 33.1%

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

      11. metadata-eval 33.1%

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

      12. +-commutative 33.1%

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

   3. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

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

      2. remove-double-neg 33.1%

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

      3. neg-mul-1 33.1%

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

      4. *-commutative 33.1%

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

      5. times-frac 33.1%

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

   5. Simplified 33.1%

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

      1. associate-*r/ 33.1%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. sub-neg 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.8%

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

      8. sqr-neg 5.8%

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

      9. sqrt-prod 5.8%

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

      10. add-sqr-sqrt 5.8%

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

      11. remove-double-neg 5.8%

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

      12. frac-2neg 5.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 33.1%

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

   7. Applied egg-rr 33.3%

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

      1. associate-/r/ 33.3%

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

      2. associate-*l/ 33.3%

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

      3. neg-mul-1 33.3%

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

      4. distribute-rgt-neg-in 33.3%

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

      5. *-lft-identity 33.3%

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

      6. distribute-neg-in 33.3%

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

      7. metadata-eval 33.3%

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

      8. unsub-neg 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around inf 99.1%

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

       1. associate-*r/ 99.1%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       2. metadata-eval 99.1%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

       3. associate-*r/ 99.1%

          \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}}{\sqrt{x}} \]

       4. metadata-eval 99.1%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}}{\sqrt{x}} \]

       5. unpow2 99.1%

          \[\leadsto \frac{\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}}{\sqrt{x}} \]

   12. Simplified 99.1%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.2%

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

   if 1 < x

   1. Initial program 33.1%

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

      1. frac-sub 33.1%

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

      2. div-inv 33.1%

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

      3. *-un-lft-identity 33.1%

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

      4. +-commutative 33.1%

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

      5. *-rgt-identity 33.1%

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

      6. metadata-eval 33.1%

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

      7. frac-times 33.1%

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

      8. un-div-inv 33.1%

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

      9. pow1/2 33.1%

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

      10. pow-flip 33.1%

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

      11. metadata-eval 33.1%

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

      12. +-commutative 33.1%

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

   3. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

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

      2. remove-double-neg 33.1%

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

      3. neg-mul-1 33.1%

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

      4. *-commutative 33.1%

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

      5. times-frac 33.1%

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

   5. Simplified 33.1%

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

      1. associate-*r/ 33.1%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. sub-neg 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.8%

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

      8. sqr-neg 5.8%

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

      9. sqrt-prod 5.8%

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

      10. add-sqr-sqrt 5.8%

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

      11. remove-double-neg 5.8%

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

      12. frac-2neg 5.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 33.1%

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

   7. Applied egg-rr 33.3%

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

      1. associate-/r/ 33.3%

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

      2. associate-*l/ 33.3%

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

      3. neg-mul-1 33.3%

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

      4. distribute-rgt-neg-in 33.3%

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

      5. *-lft-identity 33.3%

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

      6. distribute-neg-in 33.3%

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

      7. metadata-eval 33.3%

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

      8. unsub-neg 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around inf 98.6%

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

4. Final simplification 97.9%

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

Alternative 7: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.66:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 95.7%

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

   if 0.660000000000000031 < x

   1. Initial program 33.1%

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

      1. frac-sub 33.1%

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

      2. div-inv 33.1%

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

      3. *-un-lft-identity 33.1%

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

      4. +-commutative 33.1%

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

      5. *-rgt-identity 33.1%

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

      6. metadata-eval 33.1%

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

      7. frac-times 33.1%

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

      8. un-div-inv 33.1%

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

      9. pow1/2 33.1%

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

      10. pow-flip 33.1%

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

      11. metadata-eval 33.1%

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

      12. +-commutative 33.1%

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

   3. Applied egg-rr 33.1%

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

      1. associate-*r/ 33.1%

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

      2. remove-double-neg 33.1%

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

      3. neg-mul-1 33.1%

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

      4. *-commutative 33.1%

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

      5. times-frac 33.1%

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

   5. Simplified 33.1%

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

      1. associate-*r/ 33.1%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. sub-neg 33.1%

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

      5. distribute-neg-frac 33.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 5.8%

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

      8. sqr-neg 5.8%

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

      9. sqrt-prod 5.8%

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

      10. add-sqr-sqrt 5.8%

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

      11. remove-double-neg 5.8%

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

      12. frac-2neg 5.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 33.1%

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

   7. Applied egg-rr 33.3%

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

      1. associate-/r/ 33.3%

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

      2. associate-*l/ 33.3%

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

      3. neg-mul-1 33.3%

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

      4. distribute-rgt-neg-in 33.3%

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

      5. *-lft-identity 33.3%

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

      6. distribute-neg-in 33.3%

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

      7. metadata-eval 33.3%

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

      8. unsub-neg 33.3%

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

   9. Simplified 33.3%

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

   10. Taylor expanded in x around inf 98.6%

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

4. Final simplification 97.1%

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

Alternative 8: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 95.7%

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

   if 1 < x

   1. Initial program 33.1%

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

      1. *-un-lft-identity 33.1%

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

      2. clear-num 33.1%

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

      3. associate-/r/ 33.1%

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

      4. prod-diff 33.1%

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

      5. *-un-lft-identity 33.1%

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

      6. fma-neg 33.1%

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

      7. *-un-lft-identity 33.1%

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

      8. inv-pow 33.1%

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

      9. sqrt-pow2 24.6%

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

      10. metadata-eval 24.6%

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

      11. pow1/2 24.6%

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

      12. pow-flip 33.2%

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

      13. +-commutative 33.2%

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

      14. metadata-eval 33.2%

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

   3. Applied egg-rr 33.2%

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

      1. fma-udef 33.2%

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

      2. distribute-lft1-in 33.2%

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

      3. metadata-eval 33.2%

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

      4. mul0-lft 33.2%

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

      5. +-rgt-identity 33.2%

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

   5. Simplified 33.2%

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

      1. sqr-pow 19.4%

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

      2. fma-neg 6.0%

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

      3. metadata-eval 6.0%

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

      4. metadata-eval 6.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -{\left(1 + x\right)}^{-0.5}\right) \]

      5. +-commutative 6.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{-0.5}\right) \]

   7. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 31.7%

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

      1. unpow1/2 31.7%

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

      2. +-inverses 31.7%

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

   10. Simplified 31.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 66.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.5e+122) (sqrt (/ 1.0 x)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = sqrt((1.0 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = sqrt((1.0d0 / x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.sqrt((1.0 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.sqrt((1.0 / x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = sqrt(Float64(1.0 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = sqrt((1.0 / x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 71.3%

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

      1. *-un-lft-identity 71.3%

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

      2. clear-num 71.3%

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

      3. associate-/r/ 71.3%

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

      4. prod-diff 71.3%

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

      5. *-un-lft-identity 71.3%

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

      6. fma-neg 71.3%

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

      7. *-un-lft-identity 71.3%

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

      8. inv-pow 71.3%

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

      9. sqrt-pow2 71.5%

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

      10. metadata-eval 71.5%

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

      11. pow1/2 71.5%

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

      12. pow-flip 71.6%

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

      13. +-commutative 71.6%

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

      14. metadata-eval 71.6%

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

   3. Applied egg-rr 71.6%

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

      1. fma-udef 71.6%

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

      2. distribute-lft1-in 71.6%

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

      3. metadata-eval 71.6%

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

      4. mul0-lft 71.6%

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

      5. +-rgt-identity 71.6%

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

   5. Simplified 71.6%

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

      1. pow-to-exp 71.8%

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

      2. log1p-udef 71.8%

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

   7. Applied egg-rr 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. sub0-neg 71.8%

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

       2. distribute-neg-frac 71.8%

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

       3. metadata-eval 71.8%

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

   11. Simplified 71.8%

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

   12. Taylor expanded in x around inf 65.0%

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

   if 8.50000000000000003e122 < x

   1. Initial program 57.6%

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

      1. *-un-lft-identity 57.6%

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

      2. clear-num 57.6%

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

      3. associate-/r/ 57.6%

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

      4. prod-diff 57.6%

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

      5. *-un-lft-identity 57.6%

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

      6. fma-neg 57.6%

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

      7. *-un-lft-identity 57.6%

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

      8. inv-pow 57.6%

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

      9. sqrt-pow2 41.0%

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

      10. metadata-eval 41.0%

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

      11. pow1/2 41.0%

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

      12. pow-flip 57.6%

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

      13. +-commutative 57.6%

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

      14. metadata-eval 57.6%

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

   3. Applied egg-rr 57.6%

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

      1. fma-udef 57.6%

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

      2. distribute-lft1-in 57.6%

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

      3. metadata-eval 57.6%

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

      4. mul0-lft 57.6%

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

      5. +-rgt-identity 57.6%

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

   5. Simplified 57.6%

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

      1. sqr-pow 31.0%

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

      2. fma-neg 4.3%

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

      3. metadata-eval 4.3%

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

      4. metadata-eval 4.3%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -{\left(1 + x\right)}^{-0.5}\right) \]

      5. +-commutative 4.3%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{-0.5}\right) \]

   7. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 57.6%

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

      1. unpow1/2 57.6%

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

      2. +-inverses 57.6%

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

   10. Simplified 57.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 67.9%

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

2. Taylor expanded in x around 0 51.1%

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

3. Taylor expanded in x around inf 1.9%

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

4. Final simplification 1.9%

   \[\leadsto -1 \]

Alternative 11: 19.4% 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 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. inv-pow 67.9%

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

   9. sqrt-pow2 64.0%

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

   10. metadata-eval 64.0%

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

   11. pow1/2 64.0%

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

   12. pow-flip 68.1%

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

   13. +-commutative 68.1%

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

   14. metadata-eval 68.1%

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

3. Applied egg-rr 68.1%

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

   1. fma-udef 68.1%

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

   2. distribute-lft1-in 68.1%

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

   3. metadata-eval 68.1%

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

   4. mul0-lft 68.1%

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

   5. +-rgt-identity 68.1%

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

5. Simplified 68.1%

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

   1. sqr-pow 61.2%

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

   2. fma-neg 54.8%

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

   3. metadata-eval 54.8%

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

   4. metadata-eval 54.8%

      \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -{\left(1 + x\right)}^{-0.5}\right) \]

   5. +-commutative 54.8%

      \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{-0.5}\right) \]

7. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

8. Taylor expanded in x around inf 16.5%

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

   1. unpow1/2 16.5%

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

   2. +-inverses 16.5%

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

10. Simplified 16.5%

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

11. Final simplification 16.5%

    \[\leadsto 0 \]

Developer target: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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