2isqrt (example 3.6)

Percentage Accurate: 38.1% → 98.7%

Time: 9.6s

Alternatives: 8

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: 38.1% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5} \cdot 0.5 + \frac{-0.375}{{x}^{1.5}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* (pow x -0.5) 0.5) (/ -0.375 (pow x 1.5))) x))

C

double code(double x) {
	return ((pow(x, -0.5) * 0.5) + (-0.375 / pow(x, 1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x ** (-0.5d0)) * 0.5d0) + ((-0.375d0) / (x ** 1.5d0))) / x
end function

Java

public static double code(double x) {
	return ((Math.pow(x, -0.5) * 0.5) + (-0.375 / Math.pow(x, 1.5))) / x;
}

Python

def code(x):
	return ((math.pow(x, -0.5) * 0.5) + (-0.375 / math.pow(x, 1.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64((x ^ -0.5) * 0.5) + Float64(-0.375 / (x ^ 1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((x ^ -0.5) * 0.5) + (-0.375 / (x ^ 1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision] + N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Simplified 84.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \left(0.5 \cdot \left(1 + x \cdot 0.25\right)\right) + -0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right), \color{blue}{x}\right) \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

   3. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{{x}^{-1}}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

   4. sqrt-pow1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{\left(\frac{-1}{2}\right)}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{\frac{-1}{2}}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

   6. pow-lowering-pow.f64 98.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \frac{-3}{8}\right)\right), x\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot 0.5} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \left(\frac{-3}{8} \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

    2. sqrt-div N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \left(\frac{-3}{8} \cdot \frac{\sqrt{1}}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \left(\frac{-3}{8} \cdot \frac{1}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

    4. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \left(\frac{\frac{-3}{8}}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left(\sqrt{x \cdot \left(x \cdot x\right)}\right)\right)\right), x\right) \]

    6. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left(\sqrt{{x}^{3}}\right)\right)\right), x\right) \]

    7. sqrt-pow1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left({x}^{\left(\frac{3}{2}\right)}\right)\right)\right), x\right) \]

    8. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \left(\frac{3}{2}\right)\right)\right)\right), x\right) \]

    9. metadata-eval 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

12. Applied egg-rr 98.8%

    \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 + \color{blue}{\frac{-0.375}{{x}^{1.5}}}}{x} \]
13. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.375}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ -0.375 (pow x 1.5)) (/ 0.5 (sqrt x))) x))

C

double code(double x) {
	return ((-0.375 / pow(x, 1.5)) + (0.5 / sqrt(x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.375d0) / (x ** 1.5d0)) + (0.5d0 / sqrt(x))) / x
end function

Java

public static double code(double x) {
	return ((-0.375 / Math.pow(x, 1.5)) + (0.5 / Math.sqrt(x))) / x;
}

Python

def code(x):
	return ((-0.375 / math.pow(x, 1.5)) + (0.5 / math.sqrt(x))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-0.375 / (x ^ 1.5)) + Float64(0.5 / sqrt(x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-0.375 / (x ^ 1.5)) + (0.5 / sqrt(x))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Simplified 84.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \left(0.5 \cdot \left(1 + x \cdot 0.25\right)\right) + -0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right), \color{blue}{x}\right) \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right), \color{blue}{x}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   3. sqrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\sqrt{x}}\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\sqrt{x}}\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{x}\right)\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \frac{-3}{8}\right)\right), x\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{-3}{8} \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

   9. sqrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{-3}{8} \cdot \frac{\sqrt{1}}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{-3}{8} \cdot \frac{1}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{\frac{-3}{8}}{\sqrt{x \cdot \left(x \cdot x\right)}}\right)\right), x\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left(\sqrt{x \cdot \left(x \cdot x\right)}\right)\right)\right), x\right) \]

   13. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left(\sqrt{{x}^{3}}\right)\right)\right), x\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left({x}^{\left(\frac{3}{2}\right)}\right)\right)\right), x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left({x}^{\frac{3}{2}}\right)\right)\right), x\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \left({x}^{\left(\frac{1}{2} \cdot 3\right)}\right)\right)\right), x\right) \]

   17. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \left(\frac{1}{2} \cdot 3\right)\right)\right)\right), x\right) \]

   18. metadata-eval 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

10. Applied egg-rr 98.7%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{x}} + \frac{-0.375}{{x}^{1.5}}}{x}} \]

11. Final simplification 98.7%

    \[\leadsto \frac{\frac{-0.375}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}}}{x} \]
12. Add Preprocessing

Alternative 3: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \cdot \left(\left(1 - x\right) \cdot {x}^{-1.5}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} - \sqrt{x}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{{x}^{2}}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\sqrt{x}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{{x}^{2}}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   11. *-lowering-*.f64 83.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 83.2%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot x}{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-1}{2}} \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}\right)\right)\right) \]

   8. sqrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\sqrt{1}}{\sqrt{x}} - \sqrt{\color{blue}{x}}\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1}{\sqrt{x}} - \sqrt{x}\right)\right)\right)\right) \]

   10. /-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1}{\sqrt{x}} - \frac{\sqrt{x}}{\color{blue}{1}}\right)\right)\right)\right) \]

   11. frac-sub N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 \cdot 1 - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x} \cdot 1}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x}} \cdot 1}\right)\right)\right)\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 - x}{\sqrt{x} \cdot 1}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 - x}{\sqrt{x} \cdot \frac{1}{\color{blue}{1}}}\right)\right)\right)\right) \]

   15. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 - x}{\frac{\sqrt{x}}{\color{blue}{1}}}\right)\right)\right)\right) \]

   16. /-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1 - x}{\sqrt{x}}\right)\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right)\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right)\right)\right) \]

   19. sqrt-lowering-sqrt.f64 83.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right) \]

7. Applied egg-rr 83.2%

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

   1. inv-pow N/A

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

   2. times-frac N/A

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

   3. unpow-prod-down N/A

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

   4. inv-pow N/A

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

   5. clear-num N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{x}\right), \color{blue}{\left({\left(\frac{x}{\frac{1 - x}{\sqrt{x}}}\right)}^{-1}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left({\color{blue}{\left(\frac{x}{\frac{1 - x}{\sqrt{x}}}\right)}}^{-1}\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left({\left(x \cdot \frac{1}{\frac{1 - x}{\sqrt{x}}}\right)}^{-1}\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left({\left(x \cdot \frac{\sqrt{x}}{1 - x}\right)}^{-1}\right)\right) \]

   10. unpow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left({x}^{-1} \cdot \color{blue}{{\left(\frac{\sqrt{x}}{1 - x}\right)}^{-1}}\right)\right) \]

   11. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\frac{1}{x} \cdot {\color{blue}{\left(\frac{\sqrt{x}}{1 - x}\right)}}^{-1}\right)\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\frac{1}{x} \cdot \frac{1}{\color{blue}{\frac{\sqrt{x}}{1 - x}}}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\frac{1}{x} \cdot \frac{1 - x}{\color{blue}{\sqrt{x}}}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1 - x}{\sqrt{x}}\right)}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1 - x}}{\sqrt{x}}\right)\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right)\right) \]

   18. sqrt-lowering-sqrt.f64 97.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

9. Applied egg-rr 97.7%

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

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\frac{1 - x}{\sqrt{x}} \cdot \color{blue}{\frac{1}{x}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\frac{1}{\frac{\sqrt{x}}{1 - x}} \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

    3. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(\frac{1}{\sqrt{x}} \cdot \left(1 - x\right)\right) \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

    4. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(\frac{1}{{x}^{\frac{1}{2}}} \cdot \left(1 - x\right)\right) \cdot \frac{1}{x}\right)\right) \]

    5. pow-flip N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(1 - x\right)\right) \cdot \frac{1}{x}\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left({x}^{\frac{-1}{2}} \cdot \left(1 - x\right)\right) \cdot \frac{1}{x}\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(\left(1 - x\right) \cdot {x}^{\frac{-1}{2}}\right) \cdot \frac{\color{blue}{1}}{x}\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot \color{blue}{\left({x}^{\frac{-1}{2}} \cdot \frac{1}{x}\right)}\right)\right) \]

    9. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot \left({x}^{\frac{-1}{2}} \cdot {x}^{\color{blue}{-1}}\right)\right)\right) \]

    10. pow-prod-up N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {x}^{\color{blue}{\left(\frac{-1}{2} + -1\right)}}\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {x}^{\frac{-3}{2}}\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {x}^{\left(3 \cdot \color{blue}{\frac{-1}{2}}\right)}\right)\right) \]

    13. pow-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {\left({x}^{3}\right)}^{\color{blue}{\frac{-1}{2}}}\right)\right) \]

    14. cube-unmult N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {\left(x \cdot \left(x \cdot x\right)\right)}^{\frac{-1}{2}}\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot {\left(x \cdot \left(x \cdot x\right)\right)}^{\left(\frac{-1}{\color{blue}{2}}\right)}\right)\right) \]

    16. sqrt-pow1 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot \sqrt{{\left(x \cdot \left(x \cdot x\right)\right)}^{-1}}\right)\right) \]

    17. inv-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \left(\left(1 - x\right) \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

    18. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\left(1 - x\right), \color{blue}{\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)}\right)\right) \]

    19. --lowering--.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\sqrt{\color{blue}{\frac{1}{x \cdot \left(x \cdot x\right)}}}\right)\right)\right) \]

    20. inv-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\sqrt{{\left(x \cdot \left(x \cdot x\right)\right)}^{-1}}\right)\right)\right) \]

    21. sqrt-pow1 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({\left(x \cdot \left(x \cdot x\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

    22. cube-unmult N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({\left({x}^{3}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right)\right)\right) \]

    23. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left({\left({x}^{3}\right)}^{\frac{-1}{2}}\right)\right)\right) \]

11. Applied egg-rr 98.1%

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

Alternative 4: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \cdot \left(\frac{0}{x} - {x}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ -0.5 x) (- (/ 0.0 x) (pow x -0.5))))

C

double code(double x) {
	return (-0.5 / x) * ((0.0 / x) - pow(x, -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.5d0) / x) * ((0.0d0 / x) - (x ** (-0.5d0)))
end function

Java

public static double code(double x) {
	return (-0.5 / x) * ((0.0 / x) - Math.pow(x, -0.5));
}

Python

def code(x):
	return (-0.5 / x) * ((0.0 / x) - math.pow(x, -0.5))

Julia

function code(x)
	return Float64(Float64(-0.5 / x) * Float64(Float64(0.0 / x) - (x ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = (-0.5 / x) * ((0.0 / x) - (x ^ -0.5));
end

Wolfram

code[x_] := N[(N[(-0.5 / x), $MachinePrecision] * N[(N[(0.0 / x), $MachinePrecision] - N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} - \sqrt{x}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{{x}^{2}}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\sqrt{x}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{{x}^{2}}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   11. *-lowering-*.f64 83.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 83.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - \sqrt{x}\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 83.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(0 - \sqrt{x}\right)} \cdot \frac{-0.5}{x \cdot x} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\sqrt{x}\right)}{x}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]

10. Applied egg-rr 97.9%

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

11. Final simplification 97.9%

    \[\leadsto \frac{-0.5}{x} \cdot \left(\frac{0}{x} - {x}^{-0.5}\right) \]
12. Add Preprocessing

Alternative 5: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 \cdot \sqrt{x}}{x}}{0 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (* -0.5 (sqrt x)) x) (- 0.0 x)))

C

double code(double x) {
	return ((-0.5 * sqrt(x)) / x) / (0.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.5d0) * sqrt(x)) / x) / (0.0d0 - x)
end function

Java

public static double code(double x) {
	return ((-0.5 * Math.sqrt(x)) / x) / (0.0 - x);
}

Python

def code(x):
	return ((-0.5 * math.sqrt(x)) / x) / (0.0 - x)

Julia

function code(x)
	return Float64(Float64(Float64(-0.5 * sqrt(x)) / x) / Float64(0.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = ((-0.5 * sqrt(x)) / x) / (0.0 - x);
end

Wolfram

code[x_] := N[(N[(N[(-0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(0.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} - \sqrt{x}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{{x}^{2}}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\sqrt{x}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{{x}^{2}}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   11. *-lowering-*.f64 83.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 83.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - \sqrt{x}\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 83.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(0 - \sqrt{x}\right)} \cdot \frac{-0.5}{x \cdot x} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \frac{-1}{2}}{x}\right), \color{blue}{x}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \frac{-1}{2}\right), x\right), x\right) \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right), x\right) \]

   7. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right), x\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{-1}{2}\right)\right), x\right), x\right) \]

   9. sqrt-lowering-sqrt.f64 97.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{2}\right)\right), x\right), x\right) \]

10. Applied egg-rr 97.9%

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

11. Final simplification 97.9%

    \[\leadsto \frac{\frac{-0.5 \cdot \sqrt{x}}{x}}{0 - x} \]
12. Add Preprocessing

Alternative 6: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Simplified 84.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot \left(0.5 \cdot \left(1 + x \cdot 0.25\right)\right) + -0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right), \color{blue}{x}\right) \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x}} \]

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}, x\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\frac{1}{x}}\right)\right), x\right) \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), x\right) \]

    3. /-lowering-/.f64 97.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), x\right) \]

11. Simplified 97.9%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
12. Add Preprocessing

Alternative 7: 97.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} - \sqrt{x}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{{x}^{2}}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\sqrt{x}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{{x}^{2}}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\sqrt{x}\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   11. *-lowering-*.f64 83.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 83.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - \sqrt{x}\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 83.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(0 - \sqrt{x}\right)} \cdot \frac{-0.5}{x \cdot x} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \frac{-1}{2}}{x}\right), \color{blue}{x}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \frac{-1}{2}\right), x\right), x\right) \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right), x\right) \]

   7. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right), x\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{-1}{2}\right)\right), x\right), x\right) \]

   9. sqrt-lowering-sqrt.f64 97.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{2}\right)\right), x\right), x\right) \]

10. Applied egg-rr 97.9%

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

    1. associate-/l/ N/A

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

    2. distribute-rgt-neg-in N/A

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

    3. metadata-eval N/A

       \[\leadsto \frac{\sqrt{x} \cdot \frac{1}{2}}{x \cdot x} \]

    4. times-frac N/A

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

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{x}}{x}\right), \color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{x}\right), x\right), \left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right) \]

    7. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x}\right)\right) \]

    8. /-lowering-/.f64 97.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

12. Applied egg-rr 97.9%

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

Alternative 8: 35.2% 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 35.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]
4. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   2. /-lowering-/.f64 23.9%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 23.9%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 33.4%

      \[\leadsto 0 \]

7. Applied egg-rr 33.4%

   \[\leadsto \color{blue}{0} \]
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 2024150 
(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)))))