2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.6%

Time: 10.1s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

   1. flip-- 35.0%

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

   2. add-sqr-sqrt 35.7%

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

   3. add-sqr-sqrt 36.3%

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

   4. add-sqr-sqrt 36.3%

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

   5. hypot-1-def 36.3%

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

6. Applied egg-rr 36.3%

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

   1. associate--l+ 80.8%

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

   2. +-inverses 80.8%

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

   3. metadata-eval 80.8%

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

   4. +-commutative 80.8%

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

8. Simplified 80.8%

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

   1. *-un-lft-identity 80.8%

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

   2. associate-/l/ 80.8%

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

   3. *-commutative 80.8%

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

   4. hypot-undefine 80.8%

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

   5. metadata-eval 80.8%

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

   6. add-sqr-sqrt 80.8%

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

   7. distribute-rgt-in 80.8%

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

   8. *-un-lft-identity 80.8%

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

   9. add-sqr-sqrt 80.8%

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

   10. hypot-define 97.2%

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

10. Applied egg-rr 97.2%

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

    1. *-lft-identity 97.2%

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

    2. *-commutative 97.2%

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

    3. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

    \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{x + 1}} \]
14. Add Preprocessing

Alternative 2: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((-0.125d0) * sqrt((1.0d0 / x))) + (sqrt(x) * 0.5d0)) / x) / (x * (((0.5d0 + ((-0.125d0) / x)) / x) + 1.0d0))
end function

Java

public static double code(double x) {
	return (((-0.125 * Math.sqrt((1.0 / x))) + (Math.sqrt(x) * 0.5)) / x) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0));
}

Python

def code(x):
	return (((-0.125 * math.sqrt((1.0 / x))) + (math.sqrt(x) * 0.5)) / x) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (((-0.125 * sqrt((1.0 / x))) + (sqrt(x) * 0.5)) / x) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0));
end

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

7. Simplified 34.5%

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

8. Taylor expanded in x around inf 99.0%

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

9. Final simplification 99.0%

   \[\leadsto \frac{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot 0.5}{x}}{x \cdot \left(\frac{0.5 + \frac{-0.125}{x}}{x} + 1\right)} \]
10. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

   1. flip-- 35.0%

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

   2. add-sqr-sqrt 35.7%

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

   3. add-sqr-sqrt 36.3%

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

   4. add-sqr-sqrt 36.3%

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

   5. hypot-1-def 36.3%

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

6. Applied egg-rr 36.3%

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

   1. associate--l+ 80.8%

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

   2. +-inverses 80.8%

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

   3. metadata-eval 80.8%

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

   4. +-commutative 80.8%

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

8. Simplified 80.8%

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

   1. *-un-lft-identity 80.8%

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

   2. associate-/l/ 80.8%

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

   3. *-commutative 80.8%

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

   4. hypot-undefine 80.8%

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

   5. metadata-eval 80.8%

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

   6. add-sqr-sqrt 80.8%

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

   7. distribute-rgt-in 80.8%

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

   8. *-un-lft-identity 80.8%

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

   9. add-sqr-sqrt 80.8%

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

   10. hypot-define 97.2%

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

10. Applied egg-rr 97.2%

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

    1. *-lft-identity 97.2%

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

    2. *-commutative 97.2%

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

    3. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 4: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt((1.0d0 / x)) * 0.5d0) / (x * (((0.5d0 + (0.125d0 / x)) / x) + 1.0d0))
end function

Java

public static double code(double x) {
	return (Math.sqrt((1.0 / x)) * 0.5) / (x * (((0.5 + (0.125 / x)) / x) + 1.0));
}

Python

def code(x):
	return (math.sqrt((1.0 / x)) * 0.5) / (x * (((0.5 + (0.125 / x)) / x) + 1.0))

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) / Float64(x * Float64(Float64(Float64(0.5 + Float64(0.125 / x)) / x) + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (sqrt((1.0 / x)) * 0.5) / (x * (((0.5 + (0.125 / x)) / x) + 1.0));
end

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

7. Simplified 34.5%

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

8. Taylor expanded in x around inf 98.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{\left(-1 - \frac{0.5 + \frac{-0.125}{x}}{x}\right) \cdot \left(-x\right)} \]
9. Step-by-step derivation

   1. *-commutative 98.5%

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

   2. sub-neg 98.5%

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

   3. neg-mul-1 98.5%

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

   4. distribute-lft-in 98.5%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 32.1%

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

   7. sqr-neg 32.1%

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

   8. sqrt-unprod 31.8%

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

   9. add-sqr-sqrt 31.8%

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

   10. *-commutative 31.8%

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

   11. neg-mul-1 31.8%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 79.6%

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

   14. sqr-neg 79.6%

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

   15. sqrt-unprod 98.2%

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

   16. add-sqr-sqrt 98.5%

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

   17. add-sqr-sqrt 0.0%

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

   18. sqrt-unprod 79.5%

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

   19. sqr-neg 79.5%

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

   20. sqrt-unprod 98.4%

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

   21. add-sqr-sqrt 98.4%

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

   22. neg-mul-1 98.4%

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

   23. distribute-neg-frac2 98.4%

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

10. Applied egg-rr 98.5%

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

    1. *-commutative 98.5%

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

    2. distribute-rgt1-in 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x \cdot \left(\frac{0.5 + \frac{0.125}{x}}{x} + 1\right)} \]
14. Add Preprocessing

Alternative 5: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-0.5}}{x \cdot \left(\frac{0.5 + \frac{-0.125}{x}}{x} + 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 0.5 (pow x -0.5)) (* x (+ (/ (+ 0.5 (/ -0.125 x)) x) 1.0))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-0.5d0))) / (x * (((0.5d0 + ((-0.125d0) / x)) / x) + 1.0d0))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -0.5)) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0));
}

Python

def code(x):
	return (0.5 * math.pow(x, -0.5)) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -0.5)) / Float64(x * Float64(Float64(Float64(0.5 + Float64(-0.125 / x)) / x) + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -0.5)) / (x * (((0.5 + (-0.125 / x)) / x) + 1.0));
end

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

7. Simplified 34.5%

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

8. Taylor expanded in x around inf 98.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{\left(-1 - \frac{0.5 + \frac{-0.125}{x}}{x}\right) \cdot \left(-x\right)} \]
9. Step-by-step derivation

   1. inv-pow 98.5%

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

   2. sqrt-pow1 98.6%

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

   3. metadata-eval 98.6%

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

   4. *-un-lft-identity 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto \frac{0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}}{\left(-1 - \frac{0.5 + \frac{-0.125}{x}}{x}\right) \cdot \left(-x\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

    \[\leadsto \frac{0.5 \cdot {x}^{-0.5}}{x \cdot \left(\frac{0.5 + \frac{-0.125}{x}}{x} + 1\right)} \]
14. Add Preprocessing

Alternative 6: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt((1.0d0 / x)) * 0.5d0) / (x + (0.5d0 + (0.125d0 / x)))
end function

Java

public static double code(double x) {
	return (Math.sqrt((1.0 / x)) * 0.5) / (x + (0.5 + (0.125 / x)));
}

Python

def code(x):
	return (math.sqrt((1.0 / x)) * 0.5) / (x + (0.5 + (0.125 / x)))

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) / Float64(x + Float64(0.5 + Float64(0.125 / x))))
end

MATLAB

function tmp = code(x)
	tmp = (sqrt((1.0 / x)) * 0.5) / (x + (0.5 + (0.125 / x)));
end

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

7. Simplified 34.5%

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

8. Taylor expanded in x around inf 98.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{\left(-1 - \frac{0.5 + \frac{-0.125}{x}}{x}\right) \cdot \left(-x\right)} \]
9. Step-by-step derivation

   1. *-commutative 98.5%

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

   2. sub-neg 98.5%

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

   3. neg-mul-1 98.5%

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

   4. distribute-lft-in 98.5%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 32.1%

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

   7. sqr-neg 32.1%

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

   8. sqrt-unprod 31.8%

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

   9. add-sqr-sqrt 31.8%

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

   10. *-commutative 31.8%

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

   11. neg-mul-1 31.8%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 79.6%

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

   14. sqr-neg 79.6%

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

   15. sqrt-unprod 98.2%

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

   16. add-sqr-sqrt 98.5%

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

   17. add-sqr-sqrt 0.0%

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

   18. sqrt-unprod 79.5%

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

   19. sqr-neg 79.5%

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

   20. sqrt-unprod 98.4%

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

   21. add-sqr-sqrt 98.4%

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

   22. neg-mul-1 98.4%

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

   23. distribute-neg-frac2 98.4%

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

10. Applied egg-rr 98.5%

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

    1. *-commutative 98.5%

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

    2. associate-*l/ 98.5%

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

    3. associate-/l* 98.5%

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

    4. rem-square-sqrt 98.5%

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

    5. associate-*l/ 98.5%

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

    6. rem-square-sqrt 98.5%

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

    7. associate-/l/ 98.5%

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

    8. *-inverses 98.5%

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

    9. lft-mult-inverse 98.5%

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

    10. *-rgt-identity 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x + \left(0.5 + \frac{0.125}{x}\right)} \]
14. Add Preprocessing

Alternative 7: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. *-commutative 0.0%

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

   3. distribute-rgt-neg-in 0.0%

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

7. Simplified 34.5%

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

8. Taylor expanded in x around inf 98.5%

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

9. Taylor expanded in x around inf 98.5%

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

    1. distribute-rgt-in 98.5%

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

    2. *-lft-identity 98.5%

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

    3. associate-*r/ 98.5%

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

    4. metadata-eval 98.5%

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

    5. associate-*l/ 98.5%

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

    6. associate-/l* 98.5%

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

    7. rem-square-sqrt 98.5%

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

    8. associate-*l/ 98.5%

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

    9. rem-square-sqrt 98.5%

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

    10. associate-/l/ 98.5%

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

    11. *-inverses 98.5%

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

    12. lft-mult-inverse 98.5%

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

    13. metadata-eval 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

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

Alternative 8: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. frac-sub 34.8%

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

   2. *-un-lft-identity 34.8%

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

   3. +-commutative 34.8%

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

   4. *-rgt-identity 34.8%

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

   5. sqrt-unprod 34.8%

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

   6. +-commutative 34.8%

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

4. Applied egg-rr 34.8%

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

5. Taylor expanded in x around inf 34.1%

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

6. Taylor expanded in x around inf 98.4%

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

7. Final simplification 98.4%

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

Alternative 9: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in x around 0 5.5%

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

   1. inv-pow 98.5%

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

   2. sqrt-pow1 98.6%

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

   3. metadata-eval 98.6%

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

   4. *-un-lft-identity 98.6%

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

5. Applied egg-rr 5.5%

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

   1. *-lft-identity 98.6%

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

7. Simplified 5.5%

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

Developer target: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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