2sqrt (example 3.1)

Percentage Accurate: 6.8% → 98.7%

Time: 11.1s

Alternatives: 7

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{0.25}{x} - x \cdot 4}{\frac{0.25 - x}{\frac{0.5}{{x}^{-0.5}}}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (- (/ 0.25 x) (* x 4.0)) (/ (- 0.25 x) (/ 0.5 (pow x -0.5))))))

C

double code(double x) {
	return 1.0 / (((0.25 / x) - (x * 4.0)) / ((0.25 - x) / (0.5 / pow(x, -0.5))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((0.25d0 / x) - (x * 4.0d0)) / ((0.25d0 - x) / (0.5d0 / (x ** (-0.5d0)))))
end function

Java

public static double code(double x) {
	return 1.0 / (((0.25 / x) - (x * 4.0)) / ((0.25 - x) / (0.5 / Math.pow(x, -0.5))));
}

Python

def code(x):
	return 1.0 / (((0.25 / x) - (x * 4.0)) / ((0.25 - x) / (0.5 / math.pow(x, -0.5))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(0.25 / x) - Float64(x * 4.0)) / Float64(Float64(0.25 - x) / Float64(0.5 / (x ^ -0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((0.25 / x) - (x * 4.0)) / ((0.25 - x) / (0.5 / (x ^ -0.5))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(0.25 / x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 - x), $MachinePrecision] / N[(0.5 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 98.3%

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot \frac{1}{2}}}} \]

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

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

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

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

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

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.2%

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

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}}} \]

8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 98.4%

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

10. Simplified 98.4%

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

    1. flip-+ N/A

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

    2. fmm-def N/A

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

    3. *-commutative N/A

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

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

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

12. Applied egg-rr 98.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{0.25}{x} - x \cdot 4}{\frac{0.25 - x}{\frac{0.5}{{x}^{-0.5}}}}}} \]
13. Add Preprocessing

Alternative 2: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25 - x}{\frac{0.5}{{x}^{-0.5}}}}{\frac{0.25}{x} - x \cdot 4} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- 0.25 x) (/ 0.5 (pow x -0.5))) (- (/ 0.25 x) (* x 4.0))))

C

double code(double x) {
	return ((0.25 - x) / (0.5 / pow(x, -0.5))) / ((0.25 / x) - (x * 4.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.25d0 - x) / (0.5d0 / (x ** (-0.5d0)))) / ((0.25d0 / x) - (x * 4.0d0))
end function

Java

public static double code(double x) {
	return ((0.25 - x) / (0.5 / Math.pow(x, -0.5))) / ((0.25 / x) - (x * 4.0));
}

Python

def code(x):
	return ((0.25 - x) / (0.5 / math.pow(x, -0.5))) / ((0.25 / x) - (x * 4.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.25 - x) / Float64(0.5 / (x ^ -0.5))) / Float64(Float64(0.25 / x) - Float64(x * 4.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.25 - x) / (0.5 / (x ^ -0.5))) / ((0.25 / x) - (x * 4.0));
end

Wolfram

code[x_] := N[(N[(N[(0.25 - x), $MachinePrecision] / N[(0.5 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 / x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 98.3%

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot \frac{1}{2}}}} \]

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

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

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

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

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

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.2%

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

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}}} \]

8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 98.4%

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

10. Simplified 98.4%

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

    1. flip-+ N/A

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

    2. fmm-def N/A

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

    3. *-commutative N/A

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

    4. clear-num N/A

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

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

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

12. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{\frac{\frac{0.25 - x}{\frac{0.5}{{x}^{-0.5}}}}{\frac{0.25}{x} - x \cdot 4}} \]
13. Add Preprocessing

Alternative 3: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.25 + x}{\frac{0.5}{{x}^{-0.5}}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ (+ 0.25 x) (/ 0.5 (pow x -0.5)))))

C

double code(double x) {
	return 1.0 / ((0.25 + x) / (0.5 / pow(x, -0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((0.25d0 + x) / (0.5d0 / (x ** (-0.5d0))))
end function

Java

public static double code(double x) {
	return 1.0 / ((0.25 + x) / (0.5 / Math.pow(x, -0.5)));
}

Python

def code(x):
	return 1.0 / ((0.25 + x) / (0.5 / math.pow(x, -0.5)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(0.25 + x) / Float64(0.5 / (x ^ -0.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((0.25 + x) / (0.5 / (x ^ -0.5)));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(0.25 + x), $MachinePrecision] / N[(0.5 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 98.3%

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot \frac{1}{2}}}} \]

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

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

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

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

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

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.2%

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

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}}} \]

8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 98.4%

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

10. Simplified 98.4%

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. div-inv N/A

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

    4. sqrt-div N/A

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

    5. metadata-eval N/A

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

    6. div-inv N/A

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

    7. frac-add N/A

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

    8. *-commutative N/A

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

    9. pow1/2 N/A

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

    10. metadata-eval N/A

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

    11. pow-flip N/A

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

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

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

    13. rem-square-sqrt N/A

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

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

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

    15. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \left(\frac{1}{\color{blue}{{x}^{\frac{-1}{2}}}} \cdot \frac{1}{2}\right)\right)\right) \]

    16. associate-*l/ N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{{x}^{\frac{-1}{2}}}}\right)\right)\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{\frac{-1}{2}}}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{\frac{-1}{2}}\right)}\right)\right)\right) \]

    19. pow-lowering-pow.f64 97.9%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

12. Applied egg-rr 97.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{x + 0.25}{\frac{0.5}{{x}^{-0.5}}}}} \]

13. Final simplification 97.9%

    \[\leadsto \frac{1}{\frac{0.25 + x}{\frac{0.5}{{x}^{-0.5}}}} \]
14. Add Preprocessing

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{0.25 + x}{-0.5 \cdot \sqrt{x}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (/ (+ 0.25 x) (* -0.5 (sqrt x)))))

C

double code(double x) {
	return -1.0 / ((0.25 + x) / (-0.5 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / ((0.25d0 + x) / ((-0.5d0) * sqrt(x)))
end function

Java

public static double code(double x) {
	return -1.0 / ((0.25 + x) / (-0.5 * Math.sqrt(x)));
}

Python

def code(x):
	return -1.0 / ((0.25 + x) / (-0.5 * math.sqrt(x)))

Julia

function code(x)
	return Float64(-1.0 / Float64(Float64(0.25 + x) / Float64(-0.5 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / ((0.25 + x) / (-0.5 * sqrt(x)));
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(0.25 + x), $MachinePrecision] / N[(-0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 98.3%

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot \frac{1}{2}}}} \]

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

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

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

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

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

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.2%

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

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}}} \]

8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 98.4%

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

10. Simplified 98.4%

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. div-inv N/A

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

    4. frac-2neg N/A

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

    5. metadata-eval N/A

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

    6. sqrt-div N/A

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

    7. metadata-eval N/A

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

    8. div-inv N/A

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

    9. frac-2neg N/A

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

    10. metadata-eval N/A

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

    11. frac-add N/A

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

    12. sqr-neg N/A

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

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{x} \cdot \sqrt{x} + \frac{1}{4}}{\frac{-1}{2} \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right)\right) \]

    14. metadata-eval N/A

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

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

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

    16. rem-square-sqrt N/A

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

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

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

    18. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \left(\frac{-1}{2} \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right)\right)\right) \]

    20. neg-sub0 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(0 - \color{blue}{\sqrt{x}}\right)\right)\right)\right) \]

    21. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right)\right) \]

    22. sqrt-lowering-sqrt.f64 98.2%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \frac{1}{4}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right) \]

12. Applied egg-rr 98.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{x + 0.25}{-0.5 \cdot \left(0 - \sqrt{x}\right)}}} \]

13. Final simplification 98.2%

    \[\leadsto \frac{-1}{\frac{0.25 + x}{-0.5 \cdot \sqrt{x}}} \]
14. Add Preprocessing

Alternative 5: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

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

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

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

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

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

   3. /-lowering-/.f64 97.3%

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

5. Simplified 97.3%

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

   1. *-commutative N/A

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

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

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. pow1/2 N/A

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

   6. pow-flip N/A

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

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

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

   8. metadata-eval 97.5%

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

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Alternative 6: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

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

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

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

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

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

   3. /-lowering-/.f64 97.3%

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

5. Simplified 97.3%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

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

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

   5. sqrt-lowering-sqrt.f64 97.2%

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

7. Applied egg-rr 97.2%

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

Alternative 7: 3.8% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.0))

C

double code(double x) {
	return x * 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return x * 0.0;
}

Python

def code(x):
	return x * 0.0

Julia

function code(x)
	return Float64(x * 0.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.0;
end

Wolfram

code[x_] := N[(x * 0.0), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip-- N/A

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

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

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

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

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

   8. pow1/2 N/A

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 7.5%

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

4. Applied egg-rr 7.5%

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

   1. unpow1/2 N/A

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 7.5%

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

6. Applied egg-rr 7.5%

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

   1. pow1/2 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

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

   4. unpow1 N/A

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

   5. pow1/2 N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 7.1%

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

8. Applied egg-rr 7.1%

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

9. Taylor expanded in x around -inf

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

    1. mul-1-neg N/A

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

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

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

    3. rem-square-sqrt N/A

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

    4. unpow2 N/A

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

    5. *-commutative N/A

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

    6. +-inverses N/A

       \[\leadsto x \cdot \left(\mathsf{neg}\left(0\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto x \cdot 0 \]

    8. *-lowering-*.f64 3.8%

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{0}\right) \]

11. Simplified 3.8%

    \[\leadsto \color{blue}{x \cdot 0} \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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