2sqrt (example 3.1)

Percentage Accurate: 7.1% → 99.6%

Time: 10.4s

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: 7.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. pow1/2 N/A

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

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

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

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

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

   16. sqrt-lowering-sqrt.f64 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

   1. unpow1/2 N/A

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 99.6%

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

9. Applied egg-rr 99.6%

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

Alternative 2: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), \color{blue}{x}\right) \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{x} \cdot 0.5 + 0.0625 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)}{x}} \]

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 97.8%

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

8. Simplified 97.8%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. pow1/2 N/A

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

   5. inv-pow N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {x}^{\frac{-1}{2}} \]

   8. metadata-eval N/A

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

   9. sqrt-pow1 N/A

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

   10. inv-pow N/A

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

   11. *-commutative N/A

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

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

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

   13. inv-pow N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval 98.2%

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

10. Applied egg-rr 98.2%

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

Alternative 3: 97.5% 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 6.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), \color{blue}{x}\right) \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{x} \cdot 0.5 + 0.0625 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)}{x}} \]

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 97.8%

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

8. Simplified 97.8%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. pow1/2 N/A

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

   5. inv-pow N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {x}^{\frac{-1}{2}} \]

   8. metadata-eval N/A

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

   9. sqrt-pow1 N/A

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

   10. inv-pow N/A

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

   11. sqrt-div N/A

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

   12. metadata-eval N/A

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

   13. un-div-inv N/A

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

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

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

   15. sqrt-lowering-sqrt.f64 97.8%

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

10. Applied egg-rr 97.8%

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

Alternative 4: 3.8% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(1 - x\right)}{1 + x \cdot \left(x \cdot -0.125\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x (- 1.0 x)) (+ 1.0 (* x (* x -0.125)))))

C

double code(double x) {
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (1.0d0 - x)) / (1.0d0 + (x * (x * (-0.125d0))))
end function

Java

public static double code(double x) {
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
}

Python

def code(x):
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)))

Julia

function code(x)
	return Float64(Float64(x + Float64(1.0 - x)) / Float64(1.0 + Float64(x * Float64(x * -0.125))))
end

MATLAB

function tmp = code(x)
	tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
end

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. pow1/2 N/A

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

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

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

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

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

   16. sqrt-lowering-sqrt.f64 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

   6. *-lowering-*.f64 3.8%

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

7. Simplified 3.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot {x}^{2}\right)}\right)\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 3.8%

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

10. Simplified 3.8%

    \[\leadsto \frac{x + \left(1 - x\right)}{1 + \color{blue}{x \cdot \left(x \cdot -0.125\right)}} \]
11. Add Preprocessing

Alternative 5: 3.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(1 - x\right)}{x \cdot \left(x \cdot -0.125\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x (- 1.0 x)) (* x (* x -0.125))))

C

double code(double x) {
	return (x + (1.0 - x)) / (x * (x * -0.125));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (1.0d0 - x)) / (x * (x * (-0.125d0)))
end function

Java

public static double code(double x) {
	return (x + (1.0 - x)) / (x * (x * -0.125));
}

Python

def code(x):
	return (x + (1.0 - x)) / (x * (x * -0.125))

Julia

function code(x)
	return Float64(Float64(x + Float64(1.0 - x)) / Float64(x * Float64(x * -0.125)))
end

MATLAB

function tmp = code(x)
	tmp = (x + (1.0 - x)) / (x * (x * -0.125));
end

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. pow1/2 N/A

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

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

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

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

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

   16. sqrt-lowering-sqrt.f64 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

   6. *-lowering-*.f64 3.8%

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

7. Simplified 3.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \color{blue}{\left(\frac{-1}{8} \cdot {x}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 3.7%

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

10. Simplified 3.7%

    \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{x \cdot \left(x \cdot -0.125\right)}} \]
11. Add Preprocessing

Alternative 6: 3.3% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{-8}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -8.0 (* x x)))

C

double code(double x) {
	return -8.0 / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-8.0d0) / (x * x)
end function

Java

public static double code(double x) {
	return -8.0 / (x * x);
}

Python

def code(x):
	return -8.0 / (x * x)

Julia

function code(x)
	return Float64(-8.0 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -8.0 / (x * x);
end

Wolfram

code[x_] := N[(-8.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. pow1/2 N/A

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

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

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

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

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

   16. sqrt-lowering-sqrt.f64 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

   6. *-lowering-*.f64 3.8%

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

7. Simplified 3.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-8}{{x}^{2}}} \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(-8, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 3.4%

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

10. Simplified 3.4%

    \[\leadsto \color{blue}{\frac{-8}{x \cdot x}} \]
11. Add Preprocessing

Alternative 7: 1.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot -0.125\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x -0.125)))

C

double code(double x) {
	return x * (x * -0.125);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (-0.125d0))
end function

Java

public static double code(double x) {
	return x * (x * -0.125);
}

Python

def code(x):
	return x * (x * -0.125)

Julia

function code(x)
	return Float64(x * Float64(x * -0.125))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * -0.125);
end

Wolfram

code[x_] := N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
4. Step-by-step derivation

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 1.1%

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

5. Simplified 1.1%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

      \[\leadsto \frac{-1}{8} \cdot \left(x \cdot \color{blue}{x}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{8} \cdot x\right) \cdot \color{blue}{x} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{8} \cdot x\right)} \]

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 1.1%

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

8. Simplified 1.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.125\right)} \]
9. 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 2024161 
(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)))