2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.7%

Time: 8.9s

Alternatives: 5

Speedup: 205.0×

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. flip-- 6.4%

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

   2. div-inv 6.4%

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

   3. add-sqr-sqrt 6.9%

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

   4. +-commutative 6.9%

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

   5. add-sqr-sqrt 7.7%

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

   6. associate--l+ 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

   1. add0 99.6%

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

   2. flip-+ 99.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - 0 \cdot 0}{\sqrt{1 + x} - 0}} + \sqrt{x}} \]

   3. metadata-eval 99.6%

      \[\leadsto \frac{1}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \color{blue}{0}}{\sqrt{1 + x} - 0} + \sqrt{x}} \]

   4. fma-neg 99.6%

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

   5. metadata-eval 99.6%

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

   6. fma-define 99.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x} + 0}}{\sqrt{1 + x} - 0} + \sqrt{x}} \]

   7. add0 99.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}{\sqrt{1 + x} - 0} + \sqrt{x}} \]

   8. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{\sqrt{1 + x} - 0} + \sqrt{x}} \]

   9. *-un-lft-identity 99.5%

      \[\leadsto \frac{1}{\frac{1 + x}{\color{blue}{1 \cdot \sqrt{1 + x}} - 0} + \sqrt{x}} \]

   10. fma-neg 99.5%

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

   11. metadata-eval 99.5%

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

   12. fma-define 99.5%

       \[\leadsto \frac{1}{\frac{1 + x}{\color{blue}{1 \cdot \sqrt{1 + x} + 0}} + \sqrt{x}} \]

   13. *-un-lft-identity 99.5%

       \[\leadsto \frac{1}{\frac{1 + x}{\color{blue}{\sqrt{1 + x}} + 0} + \sqrt{x}} \]

   14. add0 99.5%

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

8. Applied egg-rr 99.5%

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

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

   3. pow1/2 99.5%

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

   4. pow1 99.5%

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

   5. pow-div 99.7%

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

   6. metadata-eval 99.7%

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

10. Applied egg-rr 99.7%

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

    1. unpow-1 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. flip-- 6.4%

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

   2. div-inv 6.4%

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

   3. add-sqr-sqrt 6.9%

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

   4. +-commutative 6.9%

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

   5. add-sqr-sqrt 7.7%

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

   6. associate--l+ 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} \]
8. Add Preprocessing

Alternative 3: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. flip-- 6.4%

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

   2. div-inv 6.4%

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

   3. add-sqr-sqrt 6.9%

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

   4. +-commutative 6.9%

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

   5. add-sqr-sqrt 7.7%

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

   6. associate--l+ 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. inv-pow 99.6%

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

   4. inv-pow 99.6%

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

   5. pow-prod-up 99.5%

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

   6. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. sqrt-pow1 99.6%

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

   2. metadata-eval 99.6%

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

   3. inv-pow 99.6%

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

   4. flip3-+ 62.2%

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

   5. associate-/r/ 62.2%

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

   6. sqrt-pow2 62.1%

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

   7. metadata-eval 62.1%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   8. sqrt-pow2 62.0%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   9. metadata-eval 62.0%

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

   10. add-sqr-sqrt 62.3%

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

   11. add-sqr-sqrt 62.1%

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

   12. associate-+r- 62.1%

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

10. Applied egg-rr 44.4%

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

11. Taylor expanded in x around inf 98.5%

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

12. Final simplification 98.5%

    \[\leadsto 0.5 \cdot \sqrt{\frac{1}{x}} \]
13. Add Preprocessing

Alternative 4: 7.0% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. flip-- 6.4%

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

   2. div-inv 6.4%

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

   3. add-sqr-sqrt 6.9%

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

   4. +-commutative 6.9%

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

   5. add-sqr-sqrt 7.7%

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

   6. associate--l+ 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 6.8%

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

8. Taylor expanded in x around inf 6.8%

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

9. Final simplification 6.8%

   \[\leadsto \frac{2}{x} \]
10. Add Preprocessing

Alternative 5: 6.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 6.7%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 6.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 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 2024034 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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