2isqrt (example 3.6)

Percentage Accurate: 38.9% → 97.9%

Time: 9.7s

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: 38.9% 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: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \cdot \left(0.5 + \frac{-0.5}{x}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

3. Taylor expanded in x around inf 83.0%

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

   1. distribute-lft-out-- 83.0%

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

5. Simplified 83.0%

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

   1. expm1-log1p-u 80.1%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]

   2. expm1-undefine 80.1%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)} - 1}}{{x}^{2}} \]

   3. inv-pow 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\color{blue}{{x}^{-1}}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

   4. sqrt-pow1 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

   5. metadata-eval 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left({x}^{\color{blue}{-0.5}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

7. Applied egg-rr 80.1%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)} - 1}}{{x}^{2}} \]
8. Step-by-step derivation

   1. expm1-define 80.1%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]

9. Simplified 80.1%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 83.0%

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

    2. *-commutative 83.0%

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

    3. associate-/l* 82.9%

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

11. Applied egg-rr 82.9%

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

13. Simplified 97.0%

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

    1. pow1 97.0%

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

    2. metadata-eval 97.0%

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

    3. sqrt-pow1 65.4%

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

    4. div-inv 65.4%

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

    5. sqrt-pow1 97.0%

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

    6. metadata-eval 97.0%

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

    7. pow-flip 98.7%

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

    8. metadata-eval 98.7%

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

15. Applied egg-rr 98.7%

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

    1. unpow1 98.7%

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

    2. *-commutative 98.7%

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

    3. *-commutative 98.7%

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

    4. associate-*l* 98.7%

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

    5. distribute-lft-in 98.7%

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

    6. metadata-eval 98.7%

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

    7. associate-*r/ 98.7%

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

    8. metadata-eval 98.7%

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

17. Simplified 98.7%

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

Alternative 2: 97.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

3. Taylor expanded in x around inf 83.0%

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

   1. distribute-lft-out-- 83.0%

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

5. Simplified 83.0%

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

6. Taylor expanded in x around inf 83.0%

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

   1. *-commutative 83.0%

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

8. Simplified 83.0%

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

   1. unpow2 83.0%

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

   2. times-frac 98.2%

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

   3. pow1/2 98.2%

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

   4. pow1 98.2%

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

   5. pow-div 98.3%

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

   6. metadata-eval 98.3%

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

10. Applied egg-rr 98.3%

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

Alternative 3: 96.4% accurate, 2.0× speedup

Math

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

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(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 37.3%

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

3. Taylor expanded in x around inf 83.0%

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

   1. distribute-lft-out-- 83.0%

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

5. Simplified 83.0%

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

   1. expm1-log1p-u 80.1%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]

   2. expm1-undefine 80.1%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)} - 1}}{{x}^{2}} \]

   3. inv-pow 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\sqrt{\color{blue}{{x}^{-1}}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

   4. sqrt-pow1 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

   5. metadata-eval 80.1%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(-0.5 \cdot \left({x}^{\color{blue}{-0.5}} - \sqrt{x}\right)\right)} - 1}{{x}^{2}} \]

7. Applied egg-rr 80.1%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)} - 1}}{{x}^{2}} \]
8. Step-by-step derivation

   1. expm1-define 80.1%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]

9. Simplified 80.1%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left({x}^{-0.5} - \sqrt{x}\right)\right)\right)}}{{x}^{2}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 83.0%

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

    2. *-commutative 83.0%

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

    3. associate-/l* 82.9%

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

11. Applied egg-rr 82.9%

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

13. Simplified 97.0%

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

14. Taylor expanded in x around inf 96.9%

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

15. Final simplification 96.9%

    \[\leadsto \frac{--0.5}{{x}^{1.5}} \]
16. 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 37.3%

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

   1. add-sqr-sqrt 18.7%

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

   2. sqrt-unprod 37.3%

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

   3. frac-times 32.4%

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

   4. metadata-eval 32.4%

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

   5. add-sqr-sqrt 29.0%

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

   6. +-commutative 29.0%

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

4. Applied egg-rr 29.0%

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

5. Taylor expanded in x around 0 5.6%

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

   1. unpow-1 5.6%

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

   2. metadata-eval 5.6%

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

   3. pow-sqr 5.6%

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

   4. rem-sqrt-square 5.6%

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

   5. metadata-eval 5.6%

      \[\leadsto \left|{x}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

   6. pow-sqr 5.6%

      \[\leadsto \left|\color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}}\right| \]

   7. fabs-sqr 5.6%

      \[\leadsto \color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}} \]

   8. pow-sqr 5.6%

      \[\leadsto \color{blue}{{x}^{\left(2 \cdot -0.25\right)}} \]

   9. metadata-eval 5.6%

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

7. Simplified 5.6%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
8. Add Preprocessing

Developer Target 1: 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 2024135 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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