2isqrt (example 3.6)

Percentage Accurate: 38.6% → 98.6%

Time: 12.1s

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: 38.6% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. frac-sub 38.7%

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

   2. *-rgt-identity 38.7%

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

   3. *-un-lft-identity 38.7%

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

   4. +-commutative 38.7%

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

   5. sqrt-unprod 38.7%

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

   6. +-commutative 38.7%

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

4. Applied egg-rr 38.7%

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

5. Taylor expanded in x around inf 38.2%

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

   1. associate-*r/ 38.2%

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

   2. metadata-eval 38.2%

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

7. Simplified 38.2%

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

   1. flip-- 38.8%

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

   2. add-sqr-sqrt 22.8%

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

   3. add-sqr-sqrt 39.3%

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

9. Applied egg-rr 39.3%

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

    1. associate--l+ 98.8%

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

    2. +-inverses 98.8%

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

    3. metadata-eval 98.8%

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

    4. +-commutative 98.8%

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

11. Simplified 98.8%

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

    1. *-un-lft-identity 98.8%

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

    2. associate-/l/ 97.8%

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

13. Applied egg-rr 97.8%

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

    1. *-lft-identity 97.8%

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

    2. associate-/r* 98.8%

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

    3. distribute-lft-in 98.8%

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

    4. *-rgt-identity 98.8%

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

    5. metadata-eval 98.8%

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

    6. associate-*r/ 98.8%

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

    7. *-commutative 98.8%

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

    8. associate-*r* 98.8%

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

    9. rgt-mult-inverse 98.8%

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

    10. metadata-eval 98.8%

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

15. Simplified 98.8%

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

16. Final simplification 98.8%

    \[\leadsto \frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{1 + x}} \]
17. Add Preprocessing

Alternative 2: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (pow x -1.5)) (+ 1.0 (/ 0.5 x))))

C

double code(double x) {
	return (0.5 * pow(x, -1.5)) / (1.0 + (0.5 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-1.5d0))) / (1.0d0 + (0.5d0 / x))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -1.5)) / (1.0 + (0.5 / x));
}

Python

def code(x):
	return (0.5 * math.pow(x, -1.5)) / (1.0 + (0.5 / x))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -1.5)) / Float64(1.0 + Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -1.5)) / (1.0 + (0.5 / x));
end

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. frac-sub 38.7%

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

   2. *-rgt-identity 38.7%

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

   3. *-un-lft-identity 38.7%

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

   4. +-commutative 38.7%

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

   5. sqrt-unprod 38.7%

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

   6. +-commutative 38.7%

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

4. Applied egg-rr 38.7%

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

5. Taylor expanded in x around inf 38.2%

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

   1. associate-*r/ 38.2%

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

   2. metadata-eval 38.2%

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

7. Simplified 38.2%

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

8. Taylor expanded in x around inf 97.9%

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

   1. *-commutative 97.9%

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

10. Simplified 97.9%

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

    1. times-frac 97.9%

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

    2. *-un-lft-identity 97.9%

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

    3. inv-pow 97.9%

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

    4. sqrt-pow1 97.9%

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

    5. metadata-eval 97.9%

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

    6. *-un-lft-identity 97.9%

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

    7. add-sqr-sqrt 97.6%

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

    8. times-frac 97.6%

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

    9. pow1/2 97.6%

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

    10. pow-flip 97.6%

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

    11. metadata-eval 97.6%

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

    12. un-div-inv 97.5%

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

    13. pow1/2 97.5%

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

    14. pow-flip 97.4%

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

    15. metadata-eval 97.4%

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

    16. cube-mult 97.5%

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

    17. pow-pow 98.2%

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

    18. metadata-eval 98.2%

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

12. Applied egg-rr 98.2%

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

    1. *-lft-identity 98.2%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \frac{0.5}{1 + \frac{0.5}{x}}} \]

    2. *-commutative 98.2%

       \[\leadsto \color{blue}{\frac{0.5}{1 + \frac{0.5}{x}} \cdot {x}^{-1.5}} \]

    3. associate-*l/ 98.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}}} \]

14. Simplified 98.2%

    \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}}} \]

15. Final simplification 98.2%

    \[\leadsto \frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}} \]
16. Add Preprocessing

Alternative 3: 97.7% 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 38.6%

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

3. Taylor expanded in x around inf 58.1%

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

   1. pow1 58.1%

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

   2. pow-flip 59.7%

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

   3. sqrt-pow1 98.1%

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

   4. metadata-eval 98.1%

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

   5. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

   1. unpow1 98.1%

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

7. Simplified 98.1%

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

8. Final simplification 98.1%

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

Alternative 4: 35.5% 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
3. Step-by-step derivation

   1. add-cube-cbrt 9.6%

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

   2. associate-*l* 9.6%

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

   3. frac-2neg 9.6%

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

   4. metadata-eval 9.6%

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

   5. div-inv 9.6%

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

   6. distribute-neg-frac2 9.6%

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

   7. prod-diff 6.4%

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

4. Applied egg-rr 6.4%

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

6. Simplified 6.5%

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

7. Taylor expanded in x around inf 36.0%

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

   1. distribute-rgt1-in 36.0%

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

   2. metadata-eval 36.0%

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

   3. mul0-lft 36.0%

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

9. Simplified 36.0%

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

10. Final simplification 36.0%

    \[\leadsto 0 \]
11. 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 2024059 
(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)))))