2isqrt (example 3.6)

Percentage Accurate: 69.1% → 99.4%

Time: 10.3s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. add-log-exp 48.6%

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

   2. *-un-lft-identity 48.6%

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

   3. log-prod 48.6%

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

   4. metadata-eval 48.6%

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

   5. add-log-exp 65.5%

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

   6. inv-pow 65.5%

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

   7. sqrt-pow2 60.6%

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

   8. +-commutative 60.6%

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

   9. metadata-eval 60.6%

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

3. Applied egg-rr 60.6%

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

   1. +-lft-identity 60.6%

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

5. Simplified 60.6%

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

   1. flip-- 60.6%

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

   2. frac-times 54.7%

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

   3. metadata-eval 54.7%

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

   4. add-sqr-sqrt 56.3%

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

   5. pow-prod-up 65.5%

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

   6. metadata-eval 65.5%

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

   7. inv-pow 65.5%

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

   8. inv-pow 65.5%

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

   9. sqrt-pow2 65.5%

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

   10. metadata-eval 65.5%

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

   11. metadata-eval 65.5%

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

   12. sqrt-pow2 65.5%

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

   13. inv-pow 65.5%

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

   14. div-inv 65.4%

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

7. Applied egg-rr 67.0%

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

   1. *-rgt-identity 67.0%

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

   2. *-rgt-identity 67.0%

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

   3. associate--l+ 89.6%

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

   4. +-inverses 89.6%

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

   5. metadata-eval 89.6%

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

   6. associate-*l* 98.8%

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

   7. associate-/r* 99.3%

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

9. Simplified 99.3%

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

    1. distribute-rgt-in 99.4%

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

    2. pow-plus 99.4%

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

    3. metadata-eval 99.4%

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

    4. pow1/2 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 5e-9)
   (/ (/ 1.0 x) (+ (* (sqrt (/ 1.0 x)) 1.5) (* (sqrt x) 2.0)))
   (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-9) {
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 5d-9) then
        tmp = (1.0d0 / x) / ((sqrt((1.0d0 / x)) * 1.5d0) + (sqrt(x) * 2.0d0))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 5e-9) {
		tmp = (1.0 / x) / ((Math.sqrt((1.0 / x)) * 1.5) + (Math.sqrt(x) * 2.0));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 5e-9:
		tmp = (1.0 / x) / ((math.sqrt((1.0 / x)) * 1.5) + (math.sqrt(x) * 2.0))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 5e-9)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(sqrt(Float64(1.0 / x)) * 1.5) + Float64(sqrt(x) * 2.0)));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-9)
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 37.9%

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

      1. add-log-exp 7.3%

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

      2. *-un-lft-identity 7.3%

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

      3. log-prod 7.3%

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

      4. metadata-eval 7.3%

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

      5. add-log-exp 37.9%

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

      6. inv-pow 37.9%

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

      7. sqrt-pow2 29.2%

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

      8. +-commutative 29.2%

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

      9. metadata-eval 29.2%

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

   3. Applied egg-rr 29.2%

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

      1. +-lft-identity 29.2%

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

   5. Simplified 29.2%

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

      1. flip-- 29.1%

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

      2. frac-times 18.6%

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

      3. metadata-eval 18.6%

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

      4. add-sqr-sqrt 21.5%

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

      5. pow-prod-up 38.1%

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

      6. metadata-eval 38.1%

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

      7. inv-pow 38.1%

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

      8. inv-pow 38.1%

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

      9. sqrt-pow2 38.1%

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

      10. metadata-eval 38.1%

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

      11. metadata-eval 38.1%

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

      12. sqrt-pow2 38.1%

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

      13. inv-pow 38.1%

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

      14. div-inv 38.1%

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

   7. Applied egg-rr 40.8%

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

      1. *-rgt-identity 40.8%

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

      2. *-rgt-identity 40.8%

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

      3. associate--l+ 81.8%

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

      4. +-inverses 81.8%

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

      5. metadata-eval 81.8%

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

      6. associate-*l* 98.5%

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

      7. associate-/r* 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 99.0%

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

       1. +-commutative 99.0%

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

       2. associate-+r+ 99.0%

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

       3. distribute-rgt-out 99.0%

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

       4. metadata-eval 99.0%

          \[\leadsto \frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} + 2 \cdot \sqrt{x}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 1.5 + 2 \cdot \sqrt{x}}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. frac-sub 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. +-commutative 99.3%

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

      5. *-rgt-identity 99.3%

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

      6. metadata-eval 99.3%

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

      7. frac-times 99.3%

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

      8. un-div-inv 99.3%

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

      9. pow1/2 99.3%

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

      10. pow-flip 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. times-frac 99.8%

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

      4. div-sub 99.7%

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

      5. *-inverses 99.7%

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

      6. /-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. sqrt-undiv 99.7%

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

      3. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 3e-9)
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))
   (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 3e-9) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 3d-9) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 3e-9) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 3e-9:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 3e-9)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 3e-9)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3e-9], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.99999999999999998e-9

   1. Initial program 37.5%

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

      1. frac-sub 37.5%

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

      2. div-inv 37.5%

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

      3. *-un-lft-identity 37.5%

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

      4. +-commutative 37.5%

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

      5. *-rgt-identity 37.5%

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

      6. metadata-eval 37.5%

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

      7. frac-times 37.5%

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

      8. un-div-inv 37.5%

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

      9. pow1/2 37.5%

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

      10. pow-flip 37.5%

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

      11. metadata-eval 37.5%

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

      12. +-commutative 37.5%

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

   3. Applied egg-rr 37.5%

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

      1. associate-*r/ 37.5%

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

      2. *-rgt-identity 37.5%

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

      3. times-frac 37.5%

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

      4. div-sub 37.4%

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

      5. *-inverses 37.4%

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

      6. /-rgt-identity 37.4%

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. associate-*r/ 99.3%

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

      4. metadata-eval 99.3%

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

      5. unpow2 99.3%

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

   8. Simplified 99.3%

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

   if 2.99999999999999998e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.8%

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

      1. frac-sub 98.8%

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

      2. div-inv 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. +-commutative 98.9%

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

      5. *-rgt-identity 98.9%

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

      6. metadata-eval 98.9%

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

      7. frac-times 98.8%

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

      8. un-div-inv 98.8%

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

      9. pow1/2 98.8%

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

      10. pow-flip 99.3%

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

      11. metadata-eval 99.3%

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

      12. +-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. times-frac 99.3%

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

      4. div-sub 99.2%

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

      5. *-inverses 99.2%

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

      6. /-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. sqrt-undiv 99.3%

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

      3. +-commutative 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (+ 1.0 x)))))
   (if (<= (+ (/ 1.0 (sqrt x)) t_0) 5e-9)
     (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))
     (+ (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = -1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + t_0) <= 5e-9) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = pow(x, -0.5) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + t_0) <= 5d-9) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    else
        tmp = (x ** (-0.5d0)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + t_0) <= 5e-9) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) + t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + t_0) <= 5e-9:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	else:
		tmp = math.pow(x, -0.5) + t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + t_0) <= 5e-9)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + t_0) <= 5e-9)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	else
		tmp = (x ^ -0.5) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], 5e-9], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 37.9%

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

      1. frac-sub 38.0%

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

      2. div-inv 38.0%

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

      3. *-un-lft-identity 38.0%

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

      4. +-commutative 38.0%

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

      5. *-rgt-identity 38.0%

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

      6. metadata-eval 38.0%

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

      7. frac-times 38.0%

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

      8. un-div-inv 38.0%

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

      9. pow1/2 38.0%

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

      10. pow-flip 38.0%

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

      11. metadata-eval 38.0%

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

      12. +-commutative 38.0%

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

   3. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.0%

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

      2. *-rgt-identity 38.0%

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

      3. times-frac 38.0%

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

      4. div-sub 37.9%

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

      5. *-inverses 37.9%

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

      6. /-rgt-identity 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. unpow2 98.9%

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

   8. Simplified 98.9%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. add-log-exp 7.6%

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

      2. *-un-lft-identity 7.6%

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

      3. log-prod 7.6%

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

      4. metadata-eval 7.6%

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

      5. add-log-exp 99.3%

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

      6. pow1/2 99.3%

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

      7. pow-flip 99.7%

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

      8. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 160000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 160000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 160000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 160000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 160000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 160000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 160000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 160000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 160000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e5

   1. Initial program 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. clear-num 99.3%

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

      3. associate-/r/ 99.3%

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

      4. prod-diff 99.3%

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

      5. *-un-lft-identity 99.3%

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

      6. fma-neg 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. inv-pow 99.3%

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

      9. sqrt-pow2 99.7%

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

      10. metadata-eval 99.7%

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

      11. pow1/2 99.7%

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

      12. pow-flip 99.7%

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

      13. +-commutative 99.7%

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

      14. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-lft1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. mul0-lft 99.7%

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

      5. +-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   if 1.6e5 < x

   1. Initial program 37.9%

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

      1. frac-sub 38.0%

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

      2. div-inv 38.0%

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

      3. *-un-lft-identity 38.0%

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

      4. +-commutative 38.0%

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

      5. *-rgt-identity 38.0%

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

      6. metadata-eval 38.0%

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

      7. frac-times 38.0%

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

      8. un-div-inv 38.0%

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

      9. pow1/2 38.0%

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

      10. pow-flip 38.0%

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

      11. metadata-eval 38.0%

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

      12. +-commutative 38.0%

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

   3. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.0%

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

      2. *-rgt-identity 38.0%

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

      3. times-frac 38.0%

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

      4. div-sub 37.9%

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

      5. *-inverses 37.9%

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

      6. /-rgt-identity 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. unpow2 98.9%

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

   8. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 160000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. inv-pow 99.5%

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

      9. sqrt-pow2 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 99.1%

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

   if 1.1000000000000001 < x

   1. Initial program 38.6%

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

      1. frac-sub 38.7%

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

      2. div-inv 38.7%

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

      3. *-un-lft-identity 38.7%

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

      4. +-commutative 38.7%

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

      5. *-rgt-identity 38.7%

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

      6. metadata-eval 38.7%

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

      7. frac-times 38.7%

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

      8. un-div-inv 38.7%

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

      9. pow1/2 38.7%

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

      10. pow-flip 38.7%

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

      11. metadata-eval 38.7%

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

      12. +-commutative 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. times-frac 38.7%

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

      4. div-sub 38.6%

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

      5. *-inverses 38.6%

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

      6. /-rgt-identity 38.6%

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

   5. Simplified 38.6%

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

   6. Taylor expanded in x around inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

      5. unpow2 98.3%

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

   8. Simplified 98.3%

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

4. Final simplification 98.6%

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

Alternative 7: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.5 * (pow(x, -0.5) / 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)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * ((x ** (-0.5d0)) / 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) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.5 * (Math.pow(x, -0.5) / x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. inv-pow 99.5%

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

      9. sqrt-pow2 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 99.1%

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

   if 1 < x

   1. Initial program 38.6%

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

      1. frac-sub 38.7%

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

      2. div-inv 38.7%

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

      3. *-un-lft-identity 38.7%

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

      4. +-commutative 38.7%

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

      5. *-rgt-identity 38.7%

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

      6. metadata-eval 38.7%

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

      7. frac-times 38.7%

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

      8. un-div-inv 38.7%

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

      9. pow1/2 38.7%

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

      10. pow-flip 38.7%

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

      11. metadata-eval 38.7%

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

      12. +-commutative 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. times-frac 38.7%

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

      4. div-sub 38.6%

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

      5. *-inverses 38.6%

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

      6. /-rgt-identity 38.6%

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

   5. Simplified 38.6%

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

   6. Taylor expanded in x around inf 37.2%

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

      1. associate-*r/ 37.2%

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

      2. metadata-eval 37.2%

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

   8. Simplified 37.2%

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

      1. expm1-log1p-u 37.2%

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

      2. expm1-udef 36.0%

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

      3. *-commutative 36.0%

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

      4. add-log-exp 36.0%

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

      5. associate--r- 36.0%

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

      6. metadata-eval 36.0%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-0.5} \cdot \log \left(e^{\color{blue}{0} + \frac{0.5}{x}}\right)\right)} - 1 \]

      7. metadata-eval 36.0%

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

      8. exp-sum 36.0%

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

      9. add-exp-log 36.0%

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

      10. *-un-lft-identity 36.0%

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

      11. add-log-exp 36.0%

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

   10. Applied egg-rr 36.0%

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

       1. expm1-def 96.1%

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

       2. expm1-log1p 96.1%

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

       3. *-commutative 96.1%

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

       4. metadata-eval 96.1%

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

       5. associate-*r/ 96.1%

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

       6. associate-*l* 96.0%

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

       7. associate-*l/ 96.2%

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

       8. *-lft-identity 96.2%

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

   12. Simplified 96.2%

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

4. Final simplification 97.5%

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

Alternative 8: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* 0.5 (/ (pow x -0.5) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * (pow(x, -0.5) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * ((x ** (-0.5d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * (Math.pow(x, -0.5) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 0.5 * (math.pow(x, -0.5) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 * Float64((x ^ -0.5) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 0.5 * ((x ^ -0.5) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. inv-pow 99.5%

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

      9. sqrt-pow2 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 98.3%

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

   if 0.680000000000000049 < x

   1. Initial program 38.6%

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

      1. frac-sub 38.7%

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

      2. div-inv 38.7%

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

      3. *-un-lft-identity 38.7%

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

      4. +-commutative 38.7%

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

      5. *-rgt-identity 38.7%

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

      6. metadata-eval 38.7%

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

      7. frac-times 38.7%

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

      8. un-div-inv 38.7%

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

      9. pow1/2 38.7%

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

      10. pow-flip 38.7%

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

      11. metadata-eval 38.7%

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

      12. +-commutative 38.7%

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

   3. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

      3. times-frac 38.7%

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

      4. div-sub 38.6%

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

      5. *-inverses 38.6%

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

      6. /-rgt-identity 38.6%

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

   5. Simplified 38.6%

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

   6. Taylor expanded in x around inf 37.2%

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

      1. associate-*r/ 37.2%

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

      2. metadata-eval 37.2%

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

   8. Simplified 37.2%

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

      1. expm1-log1p-u 37.2%

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

      2. expm1-udef 36.0%

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

      3. *-commutative 36.0%

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

      4. add-log-exp 36.0%

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

      5. associate--r- 36.0%

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

      6. metadata-eval 36.0%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-0.5} \cdot \log \left(e^{\color{blue}{0} + \frac{0.5}{x}}\right)\right)} - 1 \]

      7. metadata-eval 36.0%

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

      8. exp-sum 36.0%

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

      9. add-exp-log 36.0%

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

      10. *-un-lft-identity 36.0%

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

      11. add-log-exp 36.0%

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

   10. Applied egg-rr 36.0%

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

       1. expm1-def 96.1%

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

       2. expm1-log1p 96.1%

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

       3. *-commutative 96.1%

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

       4. metadata-eval 96.1%

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

       5. associate-*r/ 96.1%

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

       6. associate-*l* 96.0%

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

       7. associate-*l/ 96.2%

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

       8. *-lft-identity 96.2%

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

   12. Simplified 96.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x}\\ \end{array} \]

Alternative 9: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154) (pow x -0.5) (- (pow (* x x) -0.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = pow(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 <= 1.35d+154) then
        tmp = 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 <= 1.35e+154) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = -Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = math.pow(x, -0.5)
	else:
		tmp = -math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = x ^ -0.5;
	else
		tmp = Float64(-(Float64(x * x) ^ -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = x ^ -0.5;
	else
		tmp = -((x * x) ^ -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[Power[x, -0.5], $MachinePrecision], (-N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 66.0%

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

      1. frac-sub 66.0%

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

      2. div-inv 66.1%

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

      3. *-un-lft-identity 66.1%

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

      4. +-commutative 66.1%

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

      5. *-rgt-identity 66.1%

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

      6. metadata-eval 66.1%

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

      7. frac-times 66.0%

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

      8. un-div-inv 66.0%

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

      9. pow1/2 66.0%

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

      10. pow-flip 66.3%

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

      11. metadata-eval 66.3%

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

      12. +-commutative 66.3%

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

   3. Applied egg-rr 66.3%

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

      1. associate-*r/ 66.3%

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

      2. *-rgt-identity 66.3%

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

      3. times-frac 66.3%

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

      4. div-sub 66.3%

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

      5. *-inverses 66.3%

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

      6. /-rgt-identity 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around 0 61.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 64.1%

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

      1. add-log-exp 4.4%

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

      2. *-un-lft-identity 4.4%

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

      3. log-prod 4.4%

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

      4. metadata-eval 4.4%

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

      5. add-log-exp 64.1%

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

      6. inv-pow 64.1%

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

      7. sqrt-pow2 46.3%

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

      8. +-commutative 46.3%

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

      9. metadata-eval 46.3%

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

   3. Applied egg-rr 46.3%

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

      1. +-lft-identity 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 4.1%

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

      1. mul-1-neg 4.1%

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

   8. Simplified 4.1%

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

      1. inv-pow 4.1%

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

      2. sqrt-pow1 4.1%

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

      3. metadata-eval 4.1%

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

      4. sqr-pow 4.1%

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

      5. pow-prod-down 64.1%

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

      6. metadata-eval 64.1%

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

   10. Applied egg-rr 64.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 10: 2.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. add-log-exp 48.6%

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

   2. *-un-lft-identity 48.6%

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

   3. log-prod 48.6%

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

   4. metadata-eval 48.6%

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

   5. add-log-exp 65.5%

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

   6. inv-pow 65.5%

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

   7. sqrt-pow2 60.6%

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

   8. +-commutative 60.6%

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

   9. metadata-eval 60.6%

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

3. Applied egg-rr 60.6%

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

   1. +-lft-identity 60.6%

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

5. Simplified 60.6%

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

6. Taylor expanded in x around inf 2.2%

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

   1. mul-1-neg 2.2%

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

8. Simplified 2.2%

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

   1. inv-pow 2.2%

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

   2. sqrt-pow1 2.2%

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

   3. metadata-eval 2.2%

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

   4. expm1-log1p-u 2.2%

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

   5. expm1-udef 19.1%

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

10. Applied egg-rr 19.1%

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

    1. expm1-def 2.2%

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

    2. expm1-log1p 2.2%

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

12. Simplified 2.2%

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

13. Final simplification 2.2%

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

Alternative 11: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 65.5%

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

   1. frac-sub 65.5%

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

   2. div-inv 65.5%

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

   3. *-un-lft-identity 65.5%

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

   4. +-commutative 65.5%

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

   5. *-rgt-identity 65.5%

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

   6. metadata-eval 65.5%

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

   7. frac-times 65.5%

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

   8. un-div-inv 65.5%

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

   9. pow1/2 65.5%

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

   10. pow-flip 65.7%

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

   11. metadata-eval 65.7%

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

   12. +-commutative 65.7%

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

3. Applied egg-rr 65.7%

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

   1. associate-*r/ 65.7%

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

   2. *-rgt-identity 65.7%

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

   3. times-frac 65.7%

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

   4. div-sub 65.7%

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

   5. *-inverses 65.7%

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

   6. /-rgt-identity 65.7%

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

5. Simplified 65.7%

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

6. Taylor expanded in x around 0 45.7%

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

7. Final simplification 45.7%

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

Developer target: 98.9% 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 2023187 
(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)))))