2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.7%

Time: 9.1s

Alternatives: 6

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: 38.5% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t\_0} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t\_0}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 0.0)
     (/ (* -0.5 (sqrt (/ 1.0 x))) (- x))
     (/ (/ 1.0 (+ (sqrt x) t_0)) (sqrt (* x (+ x 1.0)))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = (-0.5 * sqrt((1.0 / x))) / -x;
	} else {
		tmp = (1.0 / (sqrt(x) + t_0)) / sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 0.0d0) then
        tmp = ((-0.5d0) * sqrt((1.0d0 / x))) / -x
    else
        tmp = (1.0d0 / (sqrt(x) + t_0)) / sqrt((x * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = (-0.5 * Math.sqrt((1.0 / x))) / -x;
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / Math.sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 0.0:
		tmp = (-0.5 * math.sqrt((1.0 / x))) / -x
	else:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / math.sqrt((x * (x + 1.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(x + 1.0), $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 36.6%

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

      1. frac-2neg 36.6%

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

      2. metadata-eval 36.6%

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

      3. frac-sub 36.6%

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

      4. *-un-lft-identity 36.6%

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

      5. +-commutative 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in x around -inf 0.0%

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, 0\right) + \frac{0 + -0.25 \cdot \sqrt{x}}{x}}{-x}}{-x} + 0} \]

   8. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \sqrt{\frac{1}{x}}}}{-x} + 0 \]
   9. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

   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 58.6%

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

      1. frac-sub 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. +-commutative 59.4%

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

      4. *-rgt-identity 59.4%

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

      5. sqrt-unprod 59.4%

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

      6. +-commutative 59.4%

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

   4. Applied egg-rr 59.4%

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

      1. flip-- 65.0%

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

      2. add-sqr-sqrt 78.0%

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

      3. add-sqr-sqrt 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. associate--l+ 98.8%

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

      2. +-inverses 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 99.7%

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

Alternative 2: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\sqrt{\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, 0.25, 1\right) \cdot {x}^{-1.5}, -0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)}}{x}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/
   (sqrt
    (fma
     0.5
     (* (fma x 0.25 1.0) (pow x -1.5))
     (* -0.5 (- (pow x -0.5) (sqrt x)))))
   x)
  2.0))

C

double code(double x) {
	return pow((sqrt(fma(0.5, (fma(x, 0.25, 1.0) * pow(x, -1.5)), (-0.5 * (pow(x, -0.5) - sqrt(x))))) / x), 2.0);
}

Julia

function code(x)
	return Float64(sqrt(fma(0.5, Float64(fma(x, 0.25, 1.0) * (x ^ -1.5)), Float64(-0.5 * Float64((x ^ -0.5) - sqrt(x))))) / x) ^ 2.0
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[N[(0.5 * N[(N[(x * 0.25 + 1.0), $MachinePrecision] * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], 2.0], $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 81.6%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 81.4%

      \[\leadsto \color{blue}{\sqrt{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \cdot \sqrt{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}}} \]

5. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, {x}^{-0.5} \cdot 0.5\right)}}{x} \cdot \frac{\sqrt{0.5 \cdot \left({x}^{-1.5} \cdot \mathsf{fma}\left(x, 0.25, 1\right)\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, {x}^{-0.5} \cdot 0.5\right)}}{x}} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, 0.25, 1\right) \cdot {x}^{-1.5}, -0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)}}{x}\right)}^{2}} \]
8. Add Preprocessing

Alternative 3: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around -inf 0.0%

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

7. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, 0\right) + \frac{0 + -0.25 \cdot \sqrt{x}}{x}}{-x}}{-x} + 0} \]

8. Taylor expanded in x around inf 98.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \sqrt{\frac{1}{x}}}}{-x} + 0 \]
9. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

10. Simplified 98.1%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

11. Final simplification 98.1%

    \[\leadsto \frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{-x} \]
12. Add Preprocessing

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around -inf 0.0%

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

7. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, 0\right) + \frac{0 + -0.25 \cdot \sqrt{x}}{x}}{-x}}{-x} + 0} \]

8. Taylor expanded in x around inf 98.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \sqrt{\frac{1}{x}}}}{-x} + 0 \]
9. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

10. Simplified 98.1%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]
11. Step-by-step derivation

    1. +-rgt-identity 98.1%

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

    2. frac-2neg 98.1%

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

    3. *-commutative 98.1%

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

    4. inv-pow 98.1%

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

    5. sqrt-pow1 98.1%

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

    6. metadata-eval 98.1%

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

    7. add-sqr-sqrt 0.0%

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

    8. sqrt-unprod 34.4%

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

    9. sqr-neg 34.4%

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

    10. sqrt-prod 34.1%

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

    11. add-sqr-sqrt 34.1%

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

    12. add-sqr-sqrt 0.0%

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

    13. sqrt-unprod 80.8%

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

    14. sqr-neg 80.8%

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

    15. sqrt-prod 97.8%

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

    16. add-sqr-sqrt 98.1%

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

12. Applied egg-rr 98.1%

    \[\leadsto \color{blue}{\frac{--0.5 \cdot {x}^{-0.5}}{x}} \]
13. Step-by-step derivation

    1. distribute-lft-neg-in 98.1%

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

    2. metadata-eval 98.1%

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

14. Simplified 98.1%

    \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]
15. Add Preprocessing

Alternative 5: 96.5% accurate, 2.0× speedup

Math

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

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. Step-by-step derivation

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around -inf 0.0%

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

7. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, 0\right) + \frac{0 + -0.25 \cdot \sqrt{x}}{x}}{-x}}{-x} + 0} \]

8. Taylor expanded in x around inf 98.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \sqrt{\frac{1}{x}}}}{-x} + 0 \]
9. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]

10. Simplified 98.1%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{-x} + 0 \]
11. Step-by-step derivation

    1. +-rgt-identity 98.1%

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

    2. distribute-frac-neg2 98.1%

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

    3. add-sqr-sqrt 97.8%

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

    4. sqrt-prod 80.8%

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

    5. sqr-neg 80.8%

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

    6. sqrt-unprod 0.0%

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

    7. add-sqr-sqrt 34.1%

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

    8. associate-/l* 34.1%

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

    9. sqrt-div 34.1%

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

    10. metadata-eval 34.1%

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

    11. frac-times 34.1%

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

    12. metadata-eval 34.1%

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

    13. add-sqr-sqrt 0.0%

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

    14. sqrt-unprod 80.8%

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

    15. sqr-neg 80.8%

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

    16. unpow2 80.8%

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

    17. sqrt-prod 67.2%

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

    18. unpow2 67.2%

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

    19. cube-mult 67.2%

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

    20. sqrt-pow1 96.0%

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

    21. metadata-eval 96.0%

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

12. Applied egg-rr 96.0%

    \[\leadsto \color{blue}{-\frac{-0.5}{{x}^{1.5}}} \]
13. Step-by-step derivation

    1. distribute-neg-frac 96.0%

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

    2. metadata-eval 96.0%

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

14. Simplified 96.0%

    \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
15. Add Preprocessing

Alternative 6: 35.4% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around inf 35.2%

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

   1. distribute-rgt1-in 35.2%

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

   2. metadata-eval 35.2%

      \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

   3. mul0-lft 35.2%

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

7. Simplified 35.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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