2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.9%

Time: 11.9s

Alternatives: 4

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: 37.8% 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

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

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 (* (+ (pow (+ 1.0 x) -0.5) (pow x -0.5)) (* x (+ 1.0 x))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. inv-pow 38.1%

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

      4. sqrt-pow2 38.1%

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

      5. metadata-eval 38.1%

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

      6. inv-pow 38.1%

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

      7. sqrt-pow2 38.1%

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

      8. +-commutative 38.1%

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

      9. metadata-eval 38.1%

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

      10. frac-times 20.3%

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

      11. metadata-eval 20.3%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 20.3%

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

      14. metadata-eval 20.3%

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

      15. add-sqr-sqrt 38.2%

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

      16. +-commutative 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in x around inf 66.7%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 66.7%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

      2. associate-/r* 66.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

      3. metadata-eval 66.7%

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

      4. div-inv 66.7%

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

      5. metadata-eval 66.7%

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

      6. unpow3 66.7%

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

      7. sqrt-prod 81.3%

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

      8. sqrt-unprod 97.4%

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

      9. add-sqr-sqrt 97.7%

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

      10. /-rgt-identity 97.7%

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

      11. frac-times 99.6%

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

      12. clear-num 99.6%

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

      13. metadata-eval 99.6%

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

      14. sqrt-div 99.6%

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

      15. add-sqr-sqrt 99.2%

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

      16. pow3 99.2%

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

      17. inv-pow 99.2%

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

      18. sqrt-pow1 99.2%

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

      19. metadata-eval 99.2%

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

      20. pow-pow 100.0%

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

      21. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 60.9%

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

      1. add-cbrt-cube 58.3%

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

      2. pow1/3 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

      3. add-sqr-sqrt 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      4. pow1 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      5. pow1/2 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(x + 1\right)}^{1} \cdot \color{blue}{{\left(x + 1\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

      6. pow-prod-up 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

      7. +-commutative 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\color{blue}{\left(1 + x\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

      8. metadata-eval 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 54.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 57.9%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   6. Simplified 57.9%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   8. Applied egg-rr 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-inverses 99.6%

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

      3. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. add-cbrt-cube 7.6%

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

   2. pow1/3 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   4. pow1 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   5. pow1/2 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(x + 1\right)}^{1} \cdot \color{blue}{{\left(x + 1\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. +-commutative 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\color{blue}{\left(1 + x\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 5.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 5.7%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 7.5%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

6. Simplified 7.5%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

8. Applied egg-rr 41.1%

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

   1. associate--l+ 82.2%

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

   2. +-inverses 82.2%

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

   3. metadata-eval 82.2%

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

10. Simplified 82.2%

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

    1. inv-pow 82.2%

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

    2. associate-*r* 97.8%

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

    3. unpow-prod-down 99.5%

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

    4. +-commutative 99.5%

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

    5. inv-pow 99.5%

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

    6. +-commutative 99.5%

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

12. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.5%

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

    2. *-rgt-identity 99.5%

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

    3. unpow-1 99.5%

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

    4. *-commutative 99.5%

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

    5. distribute-rgt-in 99.5%

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

    6. pow-plus 99.5%

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

    7. metadata-eval 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

    \[\leadsto \frac{\frac{1}{{x}^{0.5} + x \cdot {\left(1 + x\right)}^{-0.5}}}{1 + x} \]
16. Add Preprocessing

Alternative 3: 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 39.2%

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

   1. flip-- 39.1%

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

   2. clear-num 39.1%

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

   3. inv-pow 39.1%

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

   4. sqrt-pow2 39.1%

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

   5. metadata-eval 39.1%

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

   6. inv-pow 39.1%

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

   7. sqrt-pow2 39.1%

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

   8. +-commutative 39.1%

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

   9. metadata-eval 39.1%

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

   10. frac-times 22.1%

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

   11. metadata-eval 22.1%

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

   12. add-sqr-sqrt 18.7%

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

   13. frac-times 22.1%

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

   14. metadata-eval 22.1%

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

   15. add-sqr-sqrt 39.3%

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

   16. +-commutative 39.3%

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

4. Applied egg-rr 39.3%

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

5. Taylor expanded in x around inf 66.4%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 66.4%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

   2. associate-/r* 66.4%

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

   3. metadata-eval 66.4%

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

   4. div-inv 66.4%

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

   5. metadata-eval 66.4%

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

   6. unpow3 66.4%

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

   7. sqrt-prod 80.3%

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

   8. sqrt-unprod 95.6%

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

   9. add-sqr-sqrt 96.0%

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

   10. /-rgt-identity 96.0%

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

   11. frac-times 97.7%

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

   12. clear-num 97.7%

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

   13. metadata-eval 97.7%

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

   14. sqrt-div 97.7%

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

   15. add-sqr-sqrt 97.3%

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

   16. pow3 97.3%

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

   17. inv-pow 97.3%

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

   18. sqrt-pow1 97.4%

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

   19. metadata-eval 97.4%

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

   20. pow-pow 98.1%

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

   21. metadata-eval 98.1%

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

7. Applied egg-rr 98.1%

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

   1. *-lft-identity 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 4: 35.1% 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 39.2%

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

3. Taylor expanded in x around inf 36.5%

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

4. Final simplification 36.5%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 98.4% 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 2024040 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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