2isqrt (example 3.6)

Percentage Accurate: 70.1% → 92.3%

Time: 10.0s

Alternatives: 10

Speedup: 1.8×

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: 70.1% 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: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. clear-num 72.5%

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

   3. associate-/r/ 72.5%

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

   4. prod-diff 72.5%

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

   5. *-un-lft-identity 72.5%

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

   6. fma-neg 72.5%

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

   7. *-un-lft-identity 72.5%

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

   8. inv-pow 72.5%

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

   9. sqrt-pow2 68.1%

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

   10. metadata-eval 68.1%

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

   11. pow1/2 68.1%

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

   12. pow-flip 72.7%

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

   13. +-commutative 72.7%

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

   14. metadata-eval 72.7%

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

3. Applied egg-rr 72.7%

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

   1. fma-udef 72.7%

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

   2. distribute-lft1-in 72.7%

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

   3. metadata-eval 72.7%

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

   4. mul0-lft 72.7%

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

   5. +-rgt-identity 72.7%

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

5. Simplified 72.7%

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

   1. metadata-eval 72.7%

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

   2. pow-flip 68.1%

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

   3. +-commutative 68.1%

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

   4. pow1/2 68.1%

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

   5. flip-- 68.0%

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

   6. pow-prod-up 61.2%

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

   7. metadata-eval 61.2%

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

   8. inv-pow 61.2%

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

   9. frac-times 66.2%

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

   10. metadata-eval 66.2%

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

   11. add-sqr-sqrt 72.4%

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

   12. pow1/2 72.4%

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

   13. +-commutative 72.4%

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

   14. pow-flip 72.4%

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

   15. metadata-eval 72.4%

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

   16. +-commutative 72.4%

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

7. Applied egg-rr 72.4%

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

   1. frac-sub 73.6%

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

   2. *-un-lft-identity 73.6%

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

   3. +-commutative 73.6%

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

   4. +-commutative 73.6%

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

9. Applied egg-rr 73.6%

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

    1. *-rgt-identity 73.6%

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

    2. associate--l+ 91.7%

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

11. Simplified 91.7%

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

    1. associate-/r* 92.6%

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

    2. +-inverses 92.6%

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

    3. metadata-eval 92.6%

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

    4. +-commutative 92.6%

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

    5. div-inv 92.6%

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

    6. +-commutative 92.6%

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

13. Applied egg-rr 92.6%

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

    1. associate-*l/ 92.6%

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

    2. *-lft-identity 92.6%

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

15. Simplified 92.6%

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

16. Final simplification 92.6%

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

Alternative 2: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-15)
   (/ (/ 1.0 (+ x (* x x))) (* 2.0 (sqrt (/ 1.0 x))))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-15) {
		tmp = (1.0 / (x + (x * x))) / (2.0 * sqrt((1.0 / 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-15) then
        tmp = (1.0d0 / (x + (x * x))) / (2.0d0 * sqrt((1.0d0 / 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-15) {
		tmp = (1.0 / (x + (x * x))) / (2.0 * Math.sqrt((1.0 / 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-15:
		tmp = (1.0 / (x + (x * x))) / (2.0 * math.sqrt((1.0 / 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-15)
		tmp = Float64(Float64(1.0 / Float64(x + Float64(x * x))) / Float64(2.0 * sqrt(Float64(1.0 / 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-15)
		tmp = (1.0 / (x + (x * x))) / (2.0 * sqrt((1.0 / 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-15], N[(N[(1.0 / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.0000000000000001e-15

   1. Initial program 41.7%

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

      1. *-un-lft-identity 41.7%

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

      2. clear-num 41.7%

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

      3. associate-/r/ 41.7%

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

      4. prod-diff 41.7%

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

      5. *-un-lft-identity 41.7%

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

      6. fma-neg 41.7%

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

      7. *-un-lft-identity 41.7%

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

      8. inv-pow 41.7%

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

      9. sqrt-pow2 32.0%

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

      10. metadata-eval 32.0%

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

      11. pow1/2 32.0%

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

      12. pow-flip 41.7%

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

      13. +-commutative 41.7%

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

      14. metadata-eval 41.7%

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

   3. Applied egg-rr 41.7%

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

      1. fma-udef 41.7%

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

      2. distribute-lft1-in 41.7%

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

      3. metadata-eval 41.7%

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

      4. mul0-lft 41.7%

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

      5. +-rgt-identity 41.7%

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

   5. Simplified 41.7%

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

      1. metadata-eval 41.7%

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

      2. pow-flip 32.0%

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

      3. +-commutative 32.0%

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

      4. pow1/2 32.0%

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

      5. flip-- 31.9%

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

      6. pow-prod-up 18.0%

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

      7. metadata-eval 18.0%

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

      8. inv-pow 18.0%

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

      9. frac-times 28.8%

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

      10. metadata-eval 28.8%

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

      11. add-sqr-sqrt 42.0%

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

      12. pow1/2 42.0%

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

      13. +-commutative 42.0%

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

      14. pow-flip 42.0%

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

      15. metadata-eval 42.0%

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

      16. +-commutative 42.0%

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

   7. Applied egg-rr 42.0%

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

      1. frac-sub 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. +-commutative 44.1%

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

      4. +-commutative 44.1%

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

   9. Applied egg-rr 44.1%

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

       1. *-rgt-identity 44.1%

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

       2. associate--l+ 83.1%

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

       3. +-inverses 83.1%

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

       4. metadata-eval 83.1%

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

       5. distribute-lft-in 83.1%

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

       6. *-rgt-identity 83.1%

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

   11. Simplified 83.1%

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

   12. Taylor expanded in x around inf 82.8%

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

   if 1.0000000000000001e-15 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. clear-num 99.2%

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

      3. associate-/r/ 99.2%

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

      4. prod-diff 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. fma-neg 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. inv-pow 99.2%

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

      9. sqrt-pow2 99.5%

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

      10. metadata-eval 99.5%

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

      11. pow1/2 99.5%

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

      12. pow-flip 99.5%

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

      13. +-commutative 99.5%

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

      14. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. distribute-lft1-in 99.5%

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

      3. metadata-eval 99.5%

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

      4. mul0-lft 99.5%

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

      5. +-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. clear-num 72.5%

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

   3. associate-/r/ 72.5%

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

   4. prod-diff 72.5%

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

   5. *-un-lft-identity 72.5%

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

   6. fma-neg 72.5%

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

   7. *-un-lft-identity 72.5%

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

   8. inv-pow 72.5%

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

   9. sqrt-pow2 68.1%

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

   10. metadata-eval 68.1%

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

   11. pow1/2 68.1%

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

   12. pow-flip 72.7%

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

   13. +-commutative 72.7%

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

   14. metadata-eval 72.7%

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

3. Applied egg-rr 72.7%

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

   1. fma-udef 72.7%

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

   2. distribute-lft1-in 72.7%

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

   3. metadata-eval 72.7%

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

   4. mul0-lft 72.7%

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

   5. +-rgt-identity 72.7%

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

5. Simplified 72.7%

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

   1. metadata-eval 72.7%

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

   2. pow-flip 68.1%

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

   3. +-commutative 68.1%

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

   4. pow1/2 68.1%

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

   5. flip-- 68.0%

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

   6. pow-prod-up 61.2%

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

   7. metadata-eval 61.2%

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

   8. inv-pow 61.2%

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

   9. frac-times 66.2%

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

   10. metadata-eval 66.2%

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

   11. add-sqr-sqrt 72.4%

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

   12. pow1/2 72.4%

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

   13. +-commutative 72.4%

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

   14. pow-flip 72.4%

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

   15. metadata-eval 72.4%

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

   16. +-commutative 72.4%

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

7. Applied egg-rr 72.4%

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

   1. frac-sub 73.6%

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

   2. *-un-lft-identity 73.6%

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

   3. +-commutative 73.6%

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

   4. +-commutative 73.6%

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

9. Applied egg-rr 73.6%

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

    1. *-rgt-identity 73.6%

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

    2. associate--l+ 91.7%

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

    3. +-inverses 91.7%

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

    4. metadata-eval 91.7%

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

    5. distribute-lft-in 91.7%

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

    6. *-rgt-identity 91.7%

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

11. Simplified 91.7%

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

12. Final simplification 91.7%

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

Alternative 4: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (/ (/ 1.0 (+ x (* x x))) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (1.0 / (x + (x * x))) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = (x ** (-0.5d0)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (1.0d0 / (x + (x * x))) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (1.0 / (x + (x * x))) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = (1.0 / (x + (x * x))) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(1.0 / Float64(x + Float64(x * x))) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = (x ^ -0.5) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (1.0 / (x + (x * x))) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 91.3%

      \[\leadsto e^{\log x \cdot -0.5} - \frac{1}{\color{blue}{0.5 \cdot x + 1}} \]
   5. Step-by-step derivation

      1. pow-to-exp 98.6%

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

      2. expm1-log1p-u 91.3%

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

      3. expm1-udef 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. expm1-def 91.3%

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

      2. expm1-log1p 98.6%

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

   8. Simplified 98.6%

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

   if 1.19999999999999996 < x

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. clear-num 43.1%

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

      3. associate-/r/ 43.1%

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

      4. prod-diff 43.1%

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

      5. *-un-lft-identity 43.1%

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

      6. fma-neg 43.1%

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

      7. *-un-lft-identity 43.1%

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

      8. inv-pow 43.1%

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

      9. sqrt-pow2 33.6%

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

      10. metadata-eval 33.6%

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

      11. pow1/2 33.6%

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

      12. pow-flip 43.1%

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

      13. +-commutative 43.1%

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

      14. metadata-eval 43.1%

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

   3. Applied egg-rr 43.1%

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

      1. fma-udef 43.1%

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

      2. distribute-lft1-in 43.1%

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

      3. metadata-eval 43.1%

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

      4. mul0-lft 43.1%

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

      5. +-rgt-identity 43.1%

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

   5. Simplified 43.1%

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

      1. metadata-eval 43.1%

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

      2. pow-flip 33.6%

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

      3. +-commutative 33.6%

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

      4. pow1/2 33.6%

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

      5. flip-- 33.6%

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

      6. pow-prod-up 20.1%

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

      7. metadata-eval 20.1%

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

      8. inv-pow 20.1%

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

      9. frac-times 30.6%

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

      10. metadata-eval 30.6%

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

      11. add-sqr-sqrt 43.5%

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

      12. pow1/2 43.5%

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

      13. +-commutative 43.5%

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

      14. pow-flip 43.5%

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

      15. metadata-eval 43.5%

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

      16. +-commutative 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. frac-sub 45.9%

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

      2. *-un-lft-identity 45.9%

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

      3. +-commutative 45.9%

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

      4. +-commutative 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. *-rgt-identity 45.9%

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

       2. associate--l+ 83.7%

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

       3. +-inverses 83.7%

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

       4. metadata-eval 83.7%

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

       5. distribute-lft-in 83.7%

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

       6. *-rgt-identity 83.7%

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

   11. Simplified 83.7%

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

   12. Taylor expanded in x around inf 81.3%

       \[\leadsto \frac{\frac{1}{x + x \cdot x}}{\color{blue}{2 \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 5: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.68)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (/ (/ 1.0 (* x x)) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.68) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (1.0 / (x * x)) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.68d0) then
        tmp = (x ** (-0.5d0)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (1.0d0 / (x * x)) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.68) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (1.0 / (x * x)) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.68:
		tmp = math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = (1.0 / (x * x)) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.68)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(1.0 / Float64(x * x)) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.68)
		tmp = (x ^ -0.5) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (1.0 / (x * x)) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.68], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.67999999999999994

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 91.3%

      \[\leadsto e^{\log x \cdot -0.5} - \frac{1}{\color{blue}{0.5 \cdot x + 1}} \]
   5. Step-by-step derivation

      1. pow-to-exp 98.6%

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

      2. expm1-log1p-u 91.3%

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

      3. expm1-udef 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. expm1-def 91.3%

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

      2. expm1-log1p 98.6%

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

   8. Simplified 98.6%

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

   if 1.67999999999999994 < x

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. clear-num 43.1%

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

      3. associate-/r/ 43.1%

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

      4. prod-diff 43.1%

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

      5. *-un-lft-identity 43.1%

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

      6. fma-neg 43.1%

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

      7. *-un-lft-identity 43.1%

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

      8. inv-pow 43.1%

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

      9. sqrt-pow2 33.6%

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

      10. metadata-eval 33.6%

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

      11. pow1/2 33.6%

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

      12. pow-flip 43.1%

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

      13. +-commutative 43.1%

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

      14. metadata-eval 43.1%

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

   3. Applied egg-rr 43.1%

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

      1. fma-udef 43.1%

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

      2. distribute-lft1-in 43.1%

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

      3. metadata-eval 43.1%

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

      4. mul0-lft 43.1%

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

      5. +-rgt-identity 43.1%

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

   5. Simplified 43.1%

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

      1. metadata-eval 43.1%

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

      2. pow-flip 33.6%

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

      3. +-commutative 33.6%

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

      4. pow1/2 33.6%

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

      5. flip-- 33.6%

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

      6. pow-prod-up 20.1%

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

      7. metadata-eval 20.1%

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

      8. inv-pow 20.1%

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

      9. frac-times 30.6%

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

      10. metadata-eval 30.6%

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

      11. add-sqr-sqrt 43.5%

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

      12. pow1/2 43.5%

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

      13. +-commutative 43.5%

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

      14. pow-flip 43.5%

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

      15. metadata-eval 43.5%

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

      16. +-commutative 43.5%

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

   7. Applied egg-rr 43.5%

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

   8. Taylor expanded in x around inf 41.8%

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

   9. Taylor expanded in x around inf 81.2%

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

       1. unpow2 81.2%

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

   11. Simplified 81.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}}}{2 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 6: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.9e+23)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 5.9e+23) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.9d+23) then
        tmp = (x ** (-0.5d0)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.9e+23) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.9e+23:
		tmp = math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.9e+23)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.9e+23)
		tmp = (x ^ -0.5) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.9e+23], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.89999999999999987e23

   1. Initial program 95.2%

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

      1. inv-pow 95.2%

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

      2. pow1/2 95.2%

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

      3. pow-pow 95.5%

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

      4. add-exp-log 88.6%

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

      5. pow-exp 88.6%

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

      6. metadata-eval 88.6%

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

   3. Applied egg-rr 88.6%

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

   4. Taylor expanded in x around 0 85.2%

      \[\leadsto e^{\log x \cdot -0.5} - \frac{1}{\color{blue}{0.5 \cdot x + 1}} \]
   5. Step-by-step derivation

      1. pow-to-exp 92.0%

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

      2. expm1-log1p-u 85.3%

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

      3. expm1-udef 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. expm1-def 85.3%

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

      2. expm1-log1p 92.0%

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

   8. Simplified 92.0%

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

   if 5.89999999999999987e23 < x

   1. Initial program 43.2%

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

      1. inv-pow 43.2%

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

      2. pow1/2 43.2%

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

      3. pow-pow 32.9%

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

      4. add-exp-log 4.4%

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

      5. pow-exp 4.4%

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

      6. metadata-eval 4.4%

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

   3. Applied egg-rr 4.4%

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

   4. Taylor expanded in x around 0 5.1%

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

   5. Taylor expanded in x around inf 5.1%

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

      1. inv-pow 5.1%

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

      2. sqrt-pow1 5.1%

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

      3. metadata-eval 5.1%

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

      4. sqr-pow 5.1%

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

      5. pow-prod-down 44.0%

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

      6. metadata-eval 44.0%

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

   7. Applied egg-rr 44.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 7: 69.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0) (+ (+ (pow x -0.5) (* x 0.5)) -1.0) (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      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 98.0%

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

   if 2 < x

   1. Initial program 42.6%

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

      1. inv-pow 42.6%

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

      2. pow1/2 42.6%

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

      3. pow-pow 33.1%

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

      4. add-exp-log 6.9%

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

      5. pow-exp 6.8%

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

      6. metadata-eval 6.8%

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

   3. Applied egg-rr 6.8%

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

   4. Taylor expanded in x around 0 5.6%

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

   5. Taylor expanded in x around inf 5.6%

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

      1. inv-pow 5.6%

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

      2. sqrt-pow1 5.6%

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

      3. metadata-eval 5.6%

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

      4. sqr-pow 5.6%

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

      5. pow-prod-down 41.3%

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

      6. metadata-eval 41.3%

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

   7. Applied egg-rr 41.3%

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

4. Final simplification 70.9%

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

Alternative 8: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (+ (pow x -0.5) -1.0) (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.8d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.8:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.8)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      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 98.1%

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

   if 0.80000000000000004 < x

   1. Initial program 43.1%

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

      1. inv-pow 43.1%

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

      2. pow1/2 43.1%

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

      3. pow-pow 33.6%

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

      4. add-exp-log 7.6%

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

      5. pow-exp 7.6%

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

      6. metadata-eval 7.6%

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

   3. Applied egg-rr 7.6%

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

   4. Taylor expanded in x around 0 5.7%

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

   5. Taylor expanded in x around inf 5.7%

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

      1. inv-pow 5.7%

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

      2. sqrt-pow1 5.7%

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

      3. metadata-eval 5.7%

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

      4. sqr-pow 5.7%

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

      5. pow-prod-down 41.1%

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

      6. metadata-eval 41.1%

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

   7. Applied egg-rr 41.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 9: 51.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. inv-pow 72.5%

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

   2. pow1/2 72.5%

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

   3. pow-pow 68.1%

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

   4. add-exp-log 51.8%

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

   5. pow-exp 51.7%

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

   6. metadata-eval 51.7%

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

3. Applied egg-rr 51.7%

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

4. Taylor expanded in x around 0 50.2%

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

5. Taylor expanded in x around inf 52.4%

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

6. Final simplification 52.4%

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

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 72.5%

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

2. Taylor expanded in x around 0 52.1%

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

3. Taylor expanded in x around inf 1.9%

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

4. Final simplification 1.9%

   \[\leadsto -1 \]

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 2023252 
(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)))))