2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.6%

Time: 13.7s

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: 38.4% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. flip-- 38.5%

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

   2. div-inv 38.5%

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

   3. frac-times 22.5%

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

   4. metadata-eval 22.5%

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

   5. add-sqr-sqrt 21.7%

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

   6. frac-times 26.9%

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

   7. metadata-eval 26.9%

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

   8. add-sqr-sqrt 38.7%

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

   9. +-commutative 38.7%

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

   10. inv-pow 38.7%

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

   11. sqrt-pow2 38.7%

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

   12. metadata-eval 38.7%

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

   13. inv-pow 38.7%

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

   14. sqrt-pow2 38.7%

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

   15. +-commutative 38.7%

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

   16. metadata-eval 38.7%

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

4. Applied egg-rr 38.7%

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

   1. associate-*r/ 38.7%

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

   2. *-rgt-identity 38.7%

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

6. Simplified 38.7%

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

   1. frac-sub 40.7%

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

   2. div-inv 40.7%

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

   3. *-un-lft-identity 40.7%

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

   4. *-rgt-identity 40.7%

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

   5. metadata-eval 40.7%

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

   6. frac-times 40.7%

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

   7. un-div-inv 40.7%

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

8. Applied egg-rr 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate--l+ 82.6%

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

   3. +-inverses 82.6%

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

   4. metadata-eval 82.6%

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

   5. *-lft-identity 82.6%

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

   6. associate-/l/ 80.8%

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

   7. metadata-eval 80.8%

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

   8. +-inverses 80.8%

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

   9. associate--l+ 40.7%

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

   10. *-rgt-identity 40.7%

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

   11. times-frac 40.7%

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

   12. associate--l+ 82.4%

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

   13. +-inverses 82.4%

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

   14. metadata-eval 82.4%

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

   15. remove-double-neg 82.4%

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

   16. distribute-lft-neg-in 82.4%

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

   17. *-commutative 82.4%

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

   18. associate-*l/ 82.6%

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

   19. *-lft-identity 82.6%

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

   20. distribute-neg-frac 82.6%

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

10. Simplified 82.6%

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

    1. *-un-lft-identity 82.6%

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

    2. associate-/l/ 80.8%

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

    3. associate-/l/ 80.8%

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

    4. +-commutative 80.8%

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

12. Applied egg-rr 80.8%

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

    1. *-lft-identity 80.8%

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

    2. associate-/l/ 80.8%

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

    3. *-commutative 80.8%

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

    4. associate-/r* 82.6%

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

    5. associate-/r* 99.6%

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

    6. distribute-rgt-in 99.6%

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

    7. pow-plus 99.6%

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

    8. metadata-eval 99.6%

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

    9. unpow1/2 99.6%

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

    10. *-commutative 99.6%

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

    11. +-commutative 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 120000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 120000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 120000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 120000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 120000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 120000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 120000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 120000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 120000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e8

   1. Initial program 79.6%

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

      1. sub-neg 79.6%

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

      2. inv-pow 79.6%

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

      3. sqrt-pow2 79.6%

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

      4. metadata-eval 79.6%

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

      5. distribute-neg-frac 79.6%

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

      6. metadata-eval 79.6%

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

      7. +-commutative 79.6%

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

   4. Applied egg-rr 79.6%

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

      1. *-rgt-identity 79.6%

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

      2. cancel-sign-sub 79.6%

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

      3. distribute-lft-neg-in 79.6%

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

      4. *-rgt-identity 79.6%

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

      5. distribute-neg-frac 79.6%

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

      6. metadata-eval 79.6%

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

      7. unpow1/2 79.6%

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

      8. exp-to-pow 77.4%

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

      9. log1p-undefine 77.4%

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

      10. *-commutative 77.4%

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

      11. exp-neg 77.8%

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

      12. *-commutative 77.8%

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

      13. distribute-rgt-neg-in 77.8%

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

      14. log1p-undefine 77.8%

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

      15. metadata-eval 77.8%

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

      16. exp-to-pow 79.9%

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

   6. Simplified 79.9%

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

   if 1.2e8 < x

   1. Initial program 36.5%

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

   3. Taylor expanded in x around inf 79.4%

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

      1. div-sub 79.4%

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

      2. *-commutative 79.4%

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

      3. associate-/l* 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-/l* 79.4%

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

      6. distribute-rgt-out-- 79.4%

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

   5. Simplified 79.4%

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

   6. Taylor expanded in x around inf 99.2%

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

   7. Taylor expanded in x around inf 66.0%

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

      1. exp-to-pow 63.5%

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

      2. *-commutative 63.5%

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

      3. exp-neg 64.1%

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

      4. distribute-lft-neg-in 64.1%

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

      5. metadata-eval 64.1%

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

      6. *-commutative 64.1%

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

      7. exp-to-pow 66.7%

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

   9. Simplified 66.7%

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

       1. sqrt-pow1 99.4%

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

       2. metadata-eval 99.4%

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

       3. *-un-lft-identity 99.4%

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

       4. *-commutative 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. *-rgt-identity 99.4%

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

   13. Simplified 99.4%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 120000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% 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 38.5%

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

3. Taylor expanded in x around inf 77.7%

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

   1. div-sub 77.7%

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

   2. *-commutative 77.7%

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

   3. associate-/l* 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-/l* 77.6%

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

   6. distribute-rgt-out-- 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in x around inf 96.6%

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

7. Taylor expanded in x around inf 64.8%

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

   1. exp-to-pow 62.4%

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

   2. *-commutative 62.4%

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

   3. exp-neg 63.0%

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

   4. distribute-lft-neg-in 63.0%

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

   5. metadata-eval 63.0%

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

   6. *-commutative 63.0%

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

   7. exp-to-pow 65.4%

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

9. Simplified 65.4%

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

    1. sqrt-pow1 96.6%

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

    2. metadata-eval 96.6%

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

    3. *-un-lft-identity 96.6%

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

    4. *-commutative 96.6%

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

11. Applied egg-rr 96.6%

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

    1. *-rgt-identity 96.6%

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

13. Simplified 96.6%

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

14. Final simplification 96.6%

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

Alternative 4: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around 0 5.8%

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

   1. inv-pow 5.8%

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

   2. sqrt-pow1 5.8%

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

   3. metadata-eval 5.8%

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

   4. *-un-lft-identity 5.8%

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

5. Applied egg-rr 5.8%

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

   1. *-lft-identity 5.8%

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

7. Simplified 5.8%

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

8. Final simplification 5.8%

   \[\leadsto {x}^{-0.5} \]
9. 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 2024079 
(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)))))