2isqrt (example 3.6)

Percentage Accurate: 39.8% → 99.9%

Time: 11.6s

Alternatives: 3

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: 39.8% 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, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((1.0 / x) / (1.0 + x)) / (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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((1.0d0 / x) / (1.0d0 + x)) / ((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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((1.0 / x) / (1.0 + x)) / (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)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = ((1.0 / x) / (1.0 + x)) / (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)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(1.0 + x)) / 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)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((1.0 / x) / (1.0 + x)) / ((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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $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 38.5%

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

   3. Taylor expanded in x around inf 82.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

   9. Taylor expanded in x around inf 67.6%

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

       1. exp-to-pow 65.2%

          \[\leadsto 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]

       2. *-commutative 65.2%

          \[\leadsto 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]

       3. rec-exp 65.9%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \]

       4. mul-1-neg 65.9%

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

       5. associate-*r* 65.9%

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

       6. metadata-eval 65.9%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

       7. *-commutative 65.9%

          \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

       8. exp-to-pow 68.2%

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

   11. Simplified 68.2%

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

       1. *-un-lft-identity 68.2%

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

       2. sqrt-pow1 100.0%

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

       3. metadata-eval 100.0%

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

       4. *-commutative 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   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 62.5%

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

      1. flip-- 62.6%

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

      2. div-inv 62.6%

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

      3. frac-times 62.3%

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

      4. metadata-eval 62.3%

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

      5. add-sqr-sqrt 61.4%

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

      6. frac-times 62.0%

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

      7. metadata-eval 62.0%

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

      8. add-sqr-sqrt 63.4%

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

      9. +-commutative 63.4%

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

      10. inv-pow 63.4%

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

      11. sqrt-pow2 63.4%

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

      12. metadata-eval 63.4%

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

      13. pow1/2 63.4%

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

      14. pow-flip 63.4%

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

      15. +-commutative 63.4%

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

      16. metadata-eval 63.4%

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

   4. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

   6. Simplified 63.4%

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

      1. frac-sub 99.4%

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

      2. *-un-lft-identity 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. associate--l+ 99.4%

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

      3. +-inverses 99.4%

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

      4. metadata-eval 99.4%

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

      5. associate-/r* 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 100.0%

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

Alternative 2: 98.0% 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 40.0%

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

3. Taylor expanded in x around inf 81.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in x around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

8. Simplified 97.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

9. Taylor expanded in x around inf 67.0%

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

    1. exp-to-pow 64.8%

       \[\leadsto 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]

    2. *-commutative 64.8%

       \[\leadsto 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]

    3. rec-exp 65.4%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \]

    4. mul-1-neg 65.4%

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

    5. associate-*r* 65.4%

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

    6. metadata-eval 65.4%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

    7. *-commutative 65.4%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

    8. exp-to-pow 67.6%

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

11. Simplified 67.6%

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

    1. *-un-lft-identity 67.6%

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

    2. sqrt-pow1 97.5%

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

    3. metadata-eval 97.5%

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

    4. *-commutative 97.5%

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

13. Applied egg-rr 97.5%

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

    1. *-rgt-identity 97.5%

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

15. Simplified 97.5%

    \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]
16. Add Preprocessing

Alternative 3: 37.0% 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 40.0%

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

   1. sub-neg 40.0%

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

   2. +-commutative 40.0%

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

   3. add-cube-cbrt 13.6%

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

   4. distribute-lft-neg-in 13.6%

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

   5. fma-define 8.2%

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

4. Applied egg-rr 8.4%

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

5. Taylor expanded in x around inf 36.5%

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

   1. distribute-rgt1-in 36.5%

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

   2. metadata-eval 36.5%

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

   3. mul0-lft 36.5%

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

7. Simplified 36.5%

   \[\leadsto \color{blue}{0} \]
8. 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 2024088 
(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)))))