2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.9%

Time: 16.8s

Alternatives: 5

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-23}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{x \cdot \left(1 + x\right)} \cdot e^{-\log \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 1e-23)
   (* 0.5 (pow x -1.5))
   (*
    (/ (+ x (- 1.0 x)) (* x (+ 1.0 x)))
    (exp (- (log (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-23) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (x * (1.0 + x))) * exp(-log((pow(x, -0.5) + pow((1.0 + x), -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 1d-23) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((x + (1.0d0 - x)) / (x * (1.0d0 + x))) * exp(-log(((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 1e-23) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (x * (1.0 + x))) * Math.exp(-Math.log((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 1e-23:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = ((x + (1.0 - x)) / (x * (1.0 + x))) * math.exp(-math.log((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 1e-23)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(Float64(x + Float64(1.0 - x)) / Float64(x * Float64(1.0 + x))) * exp(Float64(-log(Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-23)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((x + (1.0 - x)) / (x * (1.0 + x))) * exp(-log(((x ^ -0.5) + ((1.0 + x) ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-23], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[Log[N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 9.9999999999999996e-24

   1. Initial program 35.8%

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

   3. Taylor expanded in x around inf 62.3%

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

      1. pow1 62.3%

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

      2. pow-flip 64.6%

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

      3. sqrt-pow1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto {\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)}^{1} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(0.5 \cdot {x}^{-1.5}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 100.0%

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

   7. Simplified 100.0%

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

   if 9.9999999999999996e-24 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 61.7%

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

      1. add-exp-log 58.4%

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

      2. log-rec 58.4%

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

      3. pow1/2 58.4%

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

      4. log-pow 58.2%

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

      5. +-commutative 58.2%

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

      6. log1p-define 58.2%

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

   4. Applied egg-rr 58.2%

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

   6. Applied egg-rr 63.8%

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

      1. frac-sub 97.3%

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

      2. *-un-lft-identity 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. *-rgt-identity 97.3%

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

      2. associate--l+ 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-23}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{x \cdot \left(1 + x\right)} \cdot e^{-\log \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{-0.75}{x} - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) - 2 \cdot \log x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (exp
    (- (- (/ (- 0.75) x) (log (* 2.0 (sqrt (/ 1.0 x))))) (* 2.0 (log x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = exp((((-0.75 / x) - log((2.0 * sqrt((1.0 / x))))) - (2.0 * log(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = exp((((-0.75d0 / x) - log((2.0d0 * sqrt((1.0d0 / x))))) - (2.0d0 * log(x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.exp((((-0.75 / x) - Math.log((2.0 * Math.sqrt((1.0 / x))))) - (2.0 * Math.log(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.exp((((-0.75 / x) - math.log((2.0 * math.sqrt((1.0 / x))))) - (2.0 * math.log(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = exp(Float64(Float64(Float64(Float64(-0.75) / x) - log(Float64(2.0 * sqrt(Float64(1.0 / x))))) - Float64(2.0 * log(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = exp((((-0.75 / x) - log((2.0 * sqrt((1.0 / x))))) - (2.0 * log(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[((-0.75) / x), $MachinePrecision] - N[Log[N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 62.1%

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

      1. pow1 62.1%

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

      2. pow-flip 64.5%

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

      3. sqrt-pow1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto {\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)}^{1} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(0.5 \cdot {x}^{-1.5}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 100.0%

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

   7. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 59.8%

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

      1. add-exp-log 59.6%

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

      2. inv-pow 59.6%

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

      3. sqrt-pow2 59.8%

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

      4. metadata-eval 59.8%

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

      5. inv-pow 59.8%

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

      6. sqrt-pow2 60.4%

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

      7. +-commutative 60.4%

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

      8. metadata-eval 60.4%

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

   4. Applied egg-rr 60.4%

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

   6. Applied egg-rr 62.0%

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

      1. unpow-1 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in x around inf 82.3%

      \[\leadsto e^{\color{blue}{\left(2 \cdot \log \left(\frac{1}{x}\right) + \frac{0.5}{{x}^{2}}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)}} \]
   10. Step-by-step derivation

       1. associate--l+ 82.3%

          \[\leadsto e^{\color{blue}{2 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)}} \]

       2. log-rec 82.3%

          \[\leadsto e^{2 \cdot \color{blue}{\left(-\log x\right)} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       3. neg-mul-1 82.3%

          \[\leadsto e^{2 \cdot \color{blue}{\left(-1 \cdot \log x\right)} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       4. associate-*r* 82.3%

          \[\leadsto e^{\color{blue}{\left(2 \cdot -1\right) \cdot \log x} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       5. metadata-eval 82.3%

          \[\leadsto e^{\color{blue}{-2} \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       6. associate-*r/ 82.3%

          \[\leadsto e^{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\frac{0.75 \cdot 1}{x}}\right)\right)} \]

       7. metadata-eval 82.3%

          \[\leadsto e^{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{\color{blue}{0.75}}{x}\right)\right)} \]

   11. Simplified 82.3%

       \[\leadsto e^{\color{blue}{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{0.75}{x}\right)\right)}} \]

   12. Taylor expanded in x around inf 81.6%

       \[\leadsto e^{\color{blue}{2 \cdot \log \left(\frac{1}{x}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 81.6%

          \[\leadsto e^{2 \cdot \log \left(\frac{1}{x}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\frac{0.75 \cdot 1}{x}}\right)} \]

       2. metadata-eval 81.6%

          \[\leadsto e^{2 \cdot \log \left(\frac{1}{x}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{\color{blue}{0.75}}{x}\right)} \]

       3. log-rec 81.6%

          \[\leadsto e^{2 \cdot \color{blue}{\left(-\log x\right)} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{0.75}{x}\right)} \]

   14. Simplified 81.6%

       \[\leadsto e^{\color{blue}{2 \cdot \left(-\log x\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{0.75}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{-0.75}{x} - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) - 2 \cdot \log x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log x \cdot -2 - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \frac{-0.75}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (exp (+ (- (* (log x) -2.0) (log (* 2.0 (sqrt (/ 1.0 x))))) (/ -0.75 x)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = exp((((log(x) * -2.0) - log((2.0 * sqrt((1.0 / x))))) + (-0.75 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = exp((((log(x) * (-2.0d0)) - log((2.0d0 * sqrt((1.0d0 / x))))) + ((-0.75d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.exp((((Math.log(x) * -2.0) - Math.log((2.0 * Math.sqrt((1.0 / x))))) + (-0.75 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.exp((((math.log(x) * -2.0) - math.log((2.0 * math.sqrt((1.0 / x))))) + (-0.75 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = exp(Float64(Float64(Float64(log(x) * -2.0) - log(Float64(2.0 * sqrt(Float64(1.0 / x))))) + Float64(-0.75 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = exp((((log(x) * -2.0) - log((2.0 * sqrt((1.0 / x))))) + (-0.75 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] - N[Log[N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.75 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 35.9%

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

   3. Taylor expanded in x around inf 62.1%

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

      1. pow1 62.1%

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

      2. pow-flip 64.5%

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

      3. sqrt-pow1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto {\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)}^{1} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(0.5 \cdot {x}^{-1.5}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 100.0%

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

   7. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 59.8%

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

      1. add-exp-log 59.6%

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

      2. inv-pow 59.6%

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

      3. sqrt-pow2 59.8%

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

      4. metadata-eval 59.8%

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

      5. inv-pow 59.8%

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

      6. sqrt-pow2 60.4%

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

      7. +-commutative 60.4%

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

      8. metadata-eval 60.4%

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

   4. Applied egg-rr 60.4%

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

   6. Applied egg-rr 62.0%

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

      1. unpow-1 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in x around inf 82.3%

      \[\leadsto e^{\color{blue}{\left(2 \cdot \log \left(\frac{1}{x}\right) + \frac{0.5}{{x}^{2}}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)}} \]
   10. Step-by-step derivation

       1. associate--l+ 82.3%

          \[\leadsto e^{\color{blue}{2 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)}} \]

       2. log-rec 82.3%

          \[\leadsto e^{2 \cdot \color{blue}{\left(-\log x\right)} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       3. neg-mul-1 82.3%

          \[\leadsto e^{2 \cdot \color{blue}{\left(-1 \cdot \log x\right)} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       4. associate-*r* 82.3%

          \[\leadsto e^{\color{blue}{\left(2 \cdot -1\right) \cdot \log x} + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       5. metadata-eval 82.3%

          \[\leadsto e^{\color{blue}{-2} \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)\right)} \]

       6. associate-*r/ 82.3%

          \[\leadsto e^{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\frac{0.75 \cdot 1}{x}}\right)\right)} \]

       7. metadata-eval 82.3%

          \[\leadsto e^{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{\color{blue}{0.75}}{x}\right)\right)} \]

   11. Simplified 82.3%

       \[\leadsto e^{\color{blue}{-2 \cdot \log x + \left(\frac{0.5}{{x}^{2}} - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + \frac{0.75}{x}\right)\right)}} \]

   12. Taylor expanded in x around inf 81.6%

       \[\leadsto e^{\color{blue}{2 \cdot \log \left(\frac{1}{x}\right) - \left(\log \left(2 \cdot \sqrt{\frac{1}{x}}\right) + 0.75 \cdot \frac{1}{x}\right)}} \]
   13. Step-by-step derivation

       1. associate--r+ 81.5%

          \[\leadsto e^{\color{blue}{\left(2 \cdot \log \left(\frac{1}{x}\right) - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) - 0.75 \cdot \frac{1}{x}}} \]

       2. associate-*r/ 81.5%

          \[\leadsto e^{\left(2 \cdot \log \left(\frac{1}{x}\right) - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) - \color{blue}{\frac{0.75 \cdot 1}{x}}} \]

       3. metadata-eval 81.5%

          \[\leadsto e^{\left(2 \cdot \log \left(\frac{1}{x}\right) - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) - \frac{\color{blue}{0.75}}{x}} \]

       4. sub-neg 81.5%

          \[\leadsto e^{\color{blue}{\left(2 \cdot \log \left(\frac{1}{x}\right) - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \left(-\frac{0.75}{x}\right)}} \]

       5. log-rec 81.5%

          \[\leadsto e^{\left(2 \cdot \color{blue}{\left(-\log x\right)} - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \left(-\frac{0.75}{x}\right)} \]

       6. neg-mul-1 81.5%

          \[\leadsto e^{\left(2 \cdot \color{blue}{\left(-1 \cdot \log x\right)} - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \left(-\frac{0.75}{x}\right)} \]

       7. associate-*r* 81.5%

          \[\leadsto e^{\left(\color{blue}{\left(2 \cdot -1\right) \cdot \log x} - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \left(-\frac{0.75}{x}\right)} \]

       8. metadata-eval 81.5%

          \[\leadsto e^{\left(\color{blue}{-2} \cdot \log x - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \left(-\frac{0.75}{x}\right)} \]

       9. distribute-neg-frac 81.5%

          \[\leadsto e^{\left(-2 \cdot \log x - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \color{blue}{\frac{-0.75}{x}}} \]

       10. metadata-eval 81.5%

           \[\leadsto e^{\left(-2 \cdot \log x - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \frac{\color{blue}{-0.75}}{x}} \]

   14. Simplified 81.5%

       \[\leadsto e^{\color{blue}{\left(-2 \cdot \log x - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \frac{-0.75}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log x \cdot -2 - \log \left(2 \cdot \sqrt{\frac{1}{x}}\right)\right) + \frac{-0.75}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 62.0%

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

   1. pow1 62.0%

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

   2. pow-flip 64.1%

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

   3. sqrt-pow1 97.0%

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

   4. metadata-eval 97.0%

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

   5. metadata-eval 97.0%

      \[\leadsto {\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)}^{1} \]

5. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{{\left(0.5 \cdot {x}^{-1.5}\right)}^{1}} \]
6. Step-by-step derivation

   1. unpow1 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 5: 5.6% 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 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. div-inv 37.6%

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

   4. frac-2neg 37.6%

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

   5. metadata-eval 37.6%

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

   6. div-inv 37.6%

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

   7. distribute-neg-frac2 37.6%

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

   8. prod-diff 37.6%

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

   9. distribute-neg-frac 37.6%

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

   10. metadata-eval 37.6%

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

   11. +-commutative 37.6%

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

4. Applied egg-rr 31.2%

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

6. Simplified 31.2%

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

7. Taylor expanded in x around 0 5.7%

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

   1. unpow1/2 5.7%

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

   2. exp-to-pow 5.7%

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

   3. log-rec 5.7%

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

   4. distribute-lft-neg-out 5.7%

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

   5. distribute-rgt-neg-in 5.7%

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

   6. metadata-eval 5.7%

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

   7. exp-to-pow 5.7%

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

9. Simplified 5.7%

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

10. Final simplification 5.7%

    \[\leadsto {x}^{-0.5} \]
11. Add Preprocessing

Developer target: 98.3% 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 2024076 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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