2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.1%

Time: 10.5s

Alternatives: 7

Speedup: 1.5×

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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{fma}\left(x, 2, 0.5\right) + x \cdot \frac{-0.125}{x \cdot x}}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ (fma x 2.0 0.5) (* x (/ -0.125 (* x x))))) (sqrt (+ 1.0 x))))

C

double code(double x) {
	return (1.0 / (fma(x, 2.0, 0.5) + (x * (-0.125 / (x * x))))) / sqrt((1.0 + x));
}

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(fma(x, 2.0, 0.5) + Float64(x * Float64(-0.125 / Float64(x * x))))) / sqrt(Float64(1.0 + x)))
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[(x * 2.0 + 0.5), $MachinePrecision] + N[(x * N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 42.4

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

7. Applied rewrites 42.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.3%

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

11. Taylor expanded in x around inf

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

    1. sub-neg N/A

       \[\leadsto \frac{\frac{1}{x \cdot \color{blue}{\left(\left(2 + \frac{1}{2} \cdot \frac{1}{x}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right)\right)}}}{\sqrt{1 + x}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(2 + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}}{\sqrt{1 + x}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{x}\right)} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    4. lower-+.f64 N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{x}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}}{\sqrt{1 + x}} \]

    5. distribute-lft-in N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(x \cdot 2 + x \cdot \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\left(x \cdot 2 + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right) \cdot x}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    7. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{\left(x \cdot 2 + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{x} \cdot x\right)}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    8. lft-mult-inverse N/A

       \[\leadsto \frac{\frac{1}{\left(x \cdot 2 + \frac{1}{2} \cdot \color{blue}{1}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{\left(x \cdot 2 + \color{blue}{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    10. lower-fma.f64 N/A

        \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right)} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right) \cdot x}}{\sqrt{1 + x}} \]

    11. distribute-neg-frac N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{2}}} \cdot x}}{\sqrt{1 + x}} \]

    12. metadata-eval N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \frac{\color{blue}{\frac{-1}{8}}}{{x}^{2}} \cdot x}}{\sqrt{1 + x}} \]

    13. associate-*l/ N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \color{blue}{\frac{\frac{-1}{8} \cdot x}{{x}^{2}}}}}{\sqrt{1 + x}} \]

    14. *-commutative N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \frac{\color{blue}{x \cdot \frac{-1}{8}}}{{x}^{2}}}}{\sqrt{1 + x}} \]

    15. associate-*r/ N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \color{blue}{x \cdot \frac{\frac{-1}{8}}{{x}^{2}}}}}{\sqrt{1 + x}} \]

    16. metadata-eval N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{8}\right)}}{{x}^{2}}}}{\sqrt{1 + x}} \]

    17. distribute-neg-frac N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right)}}}{\sqrt{1 + x}} \]

    18. lower-*.f64 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right)}}}{\sqrt{1 + x}} \]

    19. distribute-neg-frac N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{2}}}}}{\sqrt{1 + x}} \]

    20. metadata-eval N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \frac{\color{blue}{\frac{-1}{8}}}{{x}^{2}}}}{\sqrt{1 + x}} \]

    21. lower-/.f64 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \color{blue}{\frac{\frac{-1}{8}}{{x}^{2}}}}}{\sqrt{1 + x}} \]

    22. unpow2 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, \frac{1}{2}\right) + x \cdot \frac{\frac{-1}{8}}{\color{blue}{x \cdot x}}}}{\sqrt{1 + x}} \]

    23. lower-*.f64 98.9

        \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, 2, 0.5\right) + x \cdot \frac{-0.125}{\color{blue}{x \cdot x}}}}{\sqrt{1 + x}} \]

13. Applied rewrites 98.9%

    \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5\right) + x \cdot \frac{-0.125}{x \cdot x}}}}{\sqrt{1 + x}} \]
14. Add Preprocessing

Alternative 2: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 42.4

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

7. Applied rewrites 42.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.3%

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

11. Final simplification 98.3%

    \[\leadsto \frac{\frac{1}{x + \left(x + 0.5\right)}}{\sqrt{1 + x}} \]
12. Add Preprocessing

Alternative 3: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.8

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

7. Applied rewrites 96.8%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 84.8

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

9. Applied rewrites 97.2%

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

Alternative 4: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.8

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\frac{1}{2}}{x}}{\color{blue}{\sqrt{x}}} \]
9. Step-by-step derivation

   1. lower-sqrt.f64 96.6

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

10. Applied rewrites 96.6%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\sqrt{x}}} \]
11. Add Preprocessing

Alternative 5: 37.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\mathsf{fma}\left(x, 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (fma x 0.5 1.0)))

C

double code(double x) {
	return (0.5 / x) / fma(x, 0.5, 1.0);
}

Julia

function code(x)
	return Float64(Float64(0.5 / x) / fma(x, 0.5, 1.0))
end

Wolfram

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

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.8

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\frac{1}{2}}{x}}{\color{blue}{1 + \frac{1}{2} \cdot x}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2}}{x}}{\color{blue}{\frac{1}{2} \cdot x + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2}}{x}}{\color{blue}{x \cdot \frac{1}{2}} + 1} \]

   3. lower-fma.f64 39.1

      \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}} \]

10. Applied rewrites 39.1%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}} \]
11. Add Preprocessing

Alternative 6: 7.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 \cdot \left(x \cdot 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* 1.0 (* x 2.0))))

C

double code(double x) {
	return 1.0 / (1.0 * (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 * (x * 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 * (x * 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 * (x * 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 * Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 * (x * 2.0));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.1%

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

4. Applied rewrites 43.7%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 42.4

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

7. Applied rewrites 42.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.3%

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

    1. lift-+.f64 N/A

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

    2. lift-sqrt.f64 N/A

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

    3. associate-/l/ N/A

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

    4. associate--l+ N/A

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

    5. +-inverses N/A

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

    6. metadata-eval N/A

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

    7. lift-+.f64 N/A

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

    8. lift-approx N/A

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

    9. lift-*.f64 N/A

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

    10. lower-/.f64 97.2

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

    11. lift-sqrt.f64 N/A

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

    12. lift-+.f64 N/A

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

    13. lift-approx 7.8

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

    14. lift-approx N/A

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

    15. lift-+.f64 N/A

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

    16. lift-approx 7.8

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

12. Applied rewrites 7.8%

    \[\leadsto \color{blue}{\frac{1}{1 \cdot \left(x \cdot 2\right)}} \]
13. Add Preprocessing

Alternative 7: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.7

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

5. Applied rewrites 5.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
6. Add Preprocessing

Developer Target 1: 98.2% 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]

Developer Target 2: 38.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024212 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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