2isqrt (example 3.6)

Percentage Accurate: 39.0% → 99.7%

Time: 10.0s

Alternatives: 7

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: 39.0% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t\_0} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) + \left({x}^{-2.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + x\right) - x}{\sqrt{x} + t\_0}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 2e-23)
     (/
      (+
       (fma (pow x -2.5) -0.25 (* (pow x -0.5) 0.5))
       (- (* (pow x -2.5) 0.5) (* (pow x -0.5) (/ 0.375 x))))
      x)
     (/ (/ (- (+ 1.0 x) x) (+ (sqrt x) t_0)) (sqrt (* x (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = (fma(pow(x, -2.5), -0.25, (pow(x, -0.5) * 0.5)) + ((pow(x, -2.5) * 0.5) - (pow(x, -0.5) * (0.375 / x)))) / x;
	} else {
		tmp = (((1.0 + x) - x) / (sqrt(x) + t_0)) / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 2e-23)
		tmp = Float64(Float64(fma((x ^ -2.5), -0.25, Float64((x ^ -0.5) * 0.5)) + Float64(Float64((x ^ -2.5) * 0.5) - Float64((x ^ -0.5) * Float64(0.375 / x)))) / x);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + x) - x) / Float64(sqrt(x) + t_0)) / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(N[(N[(N[Power[x, -2.5], $MachinePrecision] * -0.25 + N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, -2.5], $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(1.0 + 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))))) < 1.99999999999999992e-23

   1. Initial program 36.7%

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

   3. Taylor expanded in x around inf 79.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]
   5. Step-by-step derivation

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{{x}^{-5}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{{x}^{-5}} \cdot 0.5}{x}} \]
   7. Step-by-step derivation

      1. associate-+l- 99.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{{x}^{-5}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}}{x} \]

      2. sqrt-pow1 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{-5}{2}\right)}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      3. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\color{blue}{-2.5}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      4. pow1/2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      5. inv-pow 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      6. pow-pow 99.7%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, \color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{\color{blue}{-0.5}} \cdot 0.5\right) - \left(\frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      8. associate-/l* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      9. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left(\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot \frac{0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      10. inv-pow 99.7%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left({\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot \frac{0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      11. pow-pow 99.7%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left(\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot \frac{0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      12. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left({x}^{\color{blue}{-0.5}} \cdot \frac{0.375}{x} - \sqrt{{x}^{-5}} \cdot 0.5\right)}{x} \]

      13. sqrt-pow1 99.7%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left({x}^{-0.5} \cdot \frac{0.375}{x} - \color{blue}{{x}^{\left(\frac{-5}{2}\right)}} \cdot 0.5\right)}{x} \]

      14. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left({x}^{-0.5} \cdot \frac{0.375}{x} - {x}^{\color{blue}{-2.5}} \cdot 0.5\right)}{x} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - \left({x}^{-0.5} \cdot \frac{0.375}{x} - {x}^{-2.5} \cdot 0.5\right)}}{x} \]

   if 1.99999999999999992e-23 < (-.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 60.7%

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

      1. frac-sub 61.6%

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

      2. *-rgt-identity 61.6%

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

      3. *-un-lft-identity 61.6%

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

      4. +-commutative 61.6%

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

      5. sqrt-unprod 61.6%

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

      6. +-commutative 61.6%

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

   4. Applied egg-rr 61.6%

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

      1. flip-- 82.8%

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

      2. add-sqr-sqrt 85.2%

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

      3. +-commutative 85.2%

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

      4. add-sqr-sqrt 99.3%

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

      5. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) + \left({x}^{-2.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ t_1 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t\_1} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5 \cdot t\_0 - \frac{0.375 \cdot t\_0}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + x\right) - x}{\sqrt{x} + t\_1}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))) (t_1 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_1)) 2e-23)
     (/ (- (* 0.5 t_0) (/ (* 0.375 t_0) x)) x)
     (/ (/ (- (+ 1.0 x) x) (+ (sqrt x) t_1)) (sqrt (* x (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_1)) <= 2e-23) {
		tmp = ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
	} else {
		tmp = (((1.0 + x) - x) / (sqrt(x) + t_1)) / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    t_1 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_1)) <= 2d-23) then
        tmp = ((0.5d0 * t_0) - ((0.375d0 * t_0) / x)) / x
    else
        tmp = (((1.0d0 + x) - x) / (sqrt(x) + t_1)) / sqrt((x * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_1)) <= 2e-23) {
		tmp = ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
	} else {
		tmp = (((1.0 + x) - x) / (Math.sqrt(x) + t_1)) / Math.sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_1)) <= 2e-23:
		tmp = ((0.5 * t_0) - ((0.375 * t_0) / x)) / x
	else:
		tmp = (((1.0 + x) - x) / (math.sqrt(x) + t_1)) / math.sqrt((x * (1.0 + x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_1)) <= 2e-23)
		tmp = Float64(Float64(Float64(0.5 * t_0) - Float64(Float64(0.375 * t_0) / x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + x) - x) / Float64(sqrt(x) + t_1)) / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / x));
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_1)) <= 2e-23)
		tmp = ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
	else
		tmp = (((1.0 + x) - x) / (sqrt(x) + t_1)) / sqrt((x * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], 2e-23], N[(N[(N[(0.5 * t$95$0), $MachinePrecision] - N[(N[(0.375 * t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(1.0 + 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))))) < 1.99999999999999992e-23

   1. Initial program 36.7%

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

   3. Taylor expanded in x around inf 79.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      2. mul-1-neg 99.6%

         \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(-\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)}}{x} \]

      3. unsub-neg 99.6%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

      5. distribute-rgt-out 99.6%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.125 + 0.5\right)}}{x}}{x} \]

      6. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   if 1.99999999999999992e-23 < (-.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 60.7%

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

      1. frac-sub 61.6%

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

      2. *-rgt-identity 61.6%

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

      3. *-un-lft-identity 61.6%

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

      4. +-commutative 61.6%

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

      5. sqrt-unprod 61.6%

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

      6. +-commutative 61.6%

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

   4. Applied egg-rr 61.6%

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

      1. flip-- 82.8%

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

      2. add-sqr-sqrt 85.2%

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

      3. +-commutative 85.2%

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

      4. add-sqr-sqrt 99.3%

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

      5. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{0.375 \cdot \sqrt{\frac{1}{x}}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-13)
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-13) {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-13) then
        tmp = (0.5d0 * sqrt((1.0d0 / x))) / x
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-13) {
		tmp = (0.5 * Math.sqrt((1.0 / x))) / x;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-13:
		tmp = (0.5 * math.sqrt((1.0 / x))) / x
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-13)
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-13)
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 2.0000000000000001e-13

   1. Initial program 36.7%

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

   3. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]
   5. Step-by-step derivation

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{{x}^{-5}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{{x}^{-5}} \cdot 0.5}{x}} \]

   7. Taylor expanded in x around inf 98.9%

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

   if 2.0000000000000001e-13 < (-.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 81.4%

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

      1. sub-neg 81.4%

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

      2. inv-pow 81.4%

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

      3. sqrt-pow2 81.0%

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

      4. metadata-eval 81.0%

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

      5. distribute-neg-frac 81.0%

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

      6. metadata-eval 81.0%

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

      7. +-commutative 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. *-rgt-identity 81.0%

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

      2. cancel-sign-sub 81.0%

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

      3. distribute-lft-neg-in 81.0%

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

      4. *-rgt-identity 81.0%

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

      5. distribute-neg-frac 81.0%

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

      6. metadata-eval 81.0%

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

      7. unpow1/2 81.0%

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

      8. exp-to-pow 77.9%

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

      9. log1p-undefine 77.9%

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

      10. *-commutative 77.9%

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

      11. exp-neg 79.1%

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

      12. *-commutative 79.1%

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

      13. distribute-rgt-neg-in 79.1%

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

      14. log1p-undefine 79.1%

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

      15. metadata-eval 79.1%

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

      16. exp-to-pow 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{0.5 \cdot t\_0 - \frac{0.375 \cdot t\_0}{x}}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x)))) (/ (- (* 0.5 t_0) (/ (* 0.375 t_0) x)) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / x))
    code = ((0.5d0 * t_0) - ((0.375d0 * t_0) / x)) / x
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	return ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return ((0.5 * t_0) - ((0.375 * t_0) / x)) / x

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(0.5 * t_0) - Float64(Float64(0.375 * t_0) / x)) / x)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = ((0.5 * t_0) - ((0.375 * t_0) / x)) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.5 * t$95$0), $MachinePrecision] - N[(N[(0.375 * t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around inf 79.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 97.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. +-commutative 97.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 97.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(-\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)}}{x} \]

   3. unsub-neg 97.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. *-commutative 97.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

   5. distribute-rgt-out 97.8%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.125 + 0.5\right)}}{x}}{x} \]

   6. metadata-eval 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

   \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{0.375 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]
8. Add Preprocessing

Alternative 5: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.5e+122) (pow x -0.5) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = x ** (-0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.pow(x, -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Power[x, -0.5], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 15.8%

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

   3. Taylor expanded in x around 0 7.9%

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

      1. pow1/2 7.9%

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

      2. inv-pow 7.9%

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

      3. pow-pow 7.9%

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

      4. metadata-eval 7.9%

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

      5. *-un-lft-identity 7.9%

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

   5. Applied egg-rr 7.9%

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

      1. *-lft-identity 7.9%

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

   7. Simplified 7.9%

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

   if 8.50000000000000003e122 < x

   1. Initial program 54.0%

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

   4. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x} - \frac{1}{1 + x}, \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}, \mathsf{fma}\left(-\sqrt[3]{{\left(1 + x\right)}^{-0.5}}, \sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.5}\right)\right)} \]
   5. Step-by-step derivation

      1. fma-undefine 4.5%

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

      2. +-commutative 4.5%

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

      3. fma-undefine 19.0%

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

      4. distribute-lft-neg-out 19.0%

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

      5. *-commutative 19.0%

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

      6. distribute-lft-neg-out 19.0%

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

      7. fma-undefine 4.5%

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

      8. associate-*r/ 4.5%

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

      9. *-rgt-identity 4.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {\left(1 + x\right)}^{-0.5}\right) + \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   6. Simplified 4.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {\left(1 + x\right)}^{-0.5}\right) + \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

   7. Taylor expanded in x around inf 54.0%

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

      1. distribute-rgt1-in 54.0%

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

      2. metadata-eval 54.0%

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

      3. mul0-lft 54.0%

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

   9. Simplified 54.0%

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

Alternative 6: 97.4% accurate, 2.0× 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))) / 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 38.6%

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

3. Taylor expanded in x around inf 79.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.0%

   \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]
5. Step-by-step derivation

6. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{{x}^{-5}}, -0.25, \sqrt{\frac{1}{x}} \cdot 0.5\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{{x}^{-5}} \cdot 0.5}{x}} \]

7. Taylor expanded in x around inf 96.3%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
8. Add Preprocessing

Alternative 7: 35.8% 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 38.6%

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

4. Applied egg-rr 9.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x} - \frac{1}{1 + x}, \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}, \mathsf{fma}\left(-\sqrt[3]{{\left(1 + x\right)}^{-0.5}}, \sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.5}\right)\right)} \]
5. Step-by-step derivation

   1. fma-undefine 9.8%

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

   2. +-commutative 9.8%

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

   3. fma-undefine 18.3%

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

   4. distribute-lft-neg-out 18.3%

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

   5. *-commutative 18.3%

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

   6. distribute-lft-neg-out 18.3%

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

   7. fma-undefine 9.8%

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

   8. associate-*r/ 9.8%

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

   9. *-rgt-identity 9.8%

      \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {\left(1 + x\right)}^{-0.5}\right) + \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

6. Simplified 9.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {\left(1 + x\right)}^{-0.5}\right) + \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

7. Taylor expanded in x around inf 33.8%

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

   1. distribute-rgt1-in 33.8%

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

   2. metadata-eval 33.8%

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

   3. mul0-lft 33.8%

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

9. Simplified 33.8%

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

Developer Target 1: 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 2024136 
(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)))))