2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.2%

Time: 10.5s

Alternatives: 6

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: 38.6% 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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{\frac{1}{\sqrt{x} + t\_0}}{\sqrt{x}}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x)))) (/ (/ (/ 1.0 (+ (sqrt x) t_0)) (sqrt x)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return ((1.0 / (sqrt(x) + t_0)) / sqrt(x)) / t_0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = ((1.0d0 / (sqrt(x) + t_0)) / sqrt(x)) / t_0
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return ((1.0 / (Math.sqrt(x) + t_0)) / Math.sqrt(x)) / t_0;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return ((1.0 / (math.sqrt(x) + t_0)) / math.sqrt(x)) / t_0

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / sqrt(x)) / t_0)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = ((1.0 / (sqrt(x) + t_0)) / sqrt(x)) / t_0;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

   1. flip-- 37.8%

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

   2. add-sqr-sqrt 38.0%

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

   3. add-sqr-sqrt 39.2%

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

   4. add-sqr-sqrt 39.2%

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

   5. hypot-1-def 39.2%

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

6. Applied egg-rr 39.2%

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

   1. associate--l+ 81.8%

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

   2. +-inverses 81.8%

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

   3. metadata-eval 81.8%

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

   4. +-commutative 81.8%

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

   5. hypot-undefine 81.8%

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

   6. metadata-eval 81.8%

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

   7. rem-square-sqrt 81.8%

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

8. Simplified 81.8%

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

   1. *-un-lft-identity 81.8%

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

   2. sqrt-prod 99.2%

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

   3. times-frac 99.1%

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

10. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. associate-*l/ 99.2%

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

    3. *-lft-identity 99.2%

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

12. Simplified 99.2%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $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 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

   1. flip-- 37.8%

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

   2. add-sqr-sqrt 38.0%

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

   3. add-sqr-sqrt 39.2%

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

   4. add-sqr-sqrt 39.2%

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

   5. hypot-1-def 39.2%

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

6. Applied egg-rr 39.2%

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

   1. associate--l+ 81.8%

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

   2. +-inverses 81.8%

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

   3. metadata-eval 81.8%

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

   4. +-commutative 81.8%

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

   5. hypot-undefine 81.8%

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

   6. metadata-eval 81.8%

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

   7. rem-square-sqrt 81.8%

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

8. Simplified 81.8%

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

9. Taylor expanded in x around -inf 0.0%

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

    1. associate-*r* 0.0%

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

    2. neg-mul-1 0.0%

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

    3. +-commutative 0.0%

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

    4. unpow2 0.0%

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

    5. rem-square-sqrt 0.0%

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

    6. mul-1-neg 0.0%

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

    7. unsub-neg 0.0%

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

    8. associate-*r/ 0.0%

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

    9. metadata-eval 0.0%

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

    10. *-commutative 0.0%

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

    11. unpow2 0.0%

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

    12. rem-square-sqrt 98.7%

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

    13. neg-mul-1 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

   1. flip-- 37.8%

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

   2. add-sqr-sqrt 38.0%

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

   3. add-sqr-sqrt 39.2%

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

   4. add-sqr-sqrt 39.2%

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

   5. hypot-1-def 39.2%

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

6. Applied egg-rr 39.2%

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

   1. associate--l+ 81.8%

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

   2. +-inverses 81.8%

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

   3. metadata-eval 81.8%

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

   4. +-commutative 81.8%

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

   5. hypot-undefine 81.8%

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

   6. metadata-eval 81.8%

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

   7. rem-square-sqrt 81.8%

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

8. Simplified 81.8%

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

   1. *-un-lft-identity 81.8%

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

   2. sqrt-prod 99.2%

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

   3. times-frac 99.1%

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

10. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. associate-*l/ 99.2%

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

    3. *-lft-identity 99.2%

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

12. Simplified 99.2%

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

13. Taylor expanded in x around inf 98.1%

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

    1. associate-*r/ 98.1%

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

    2. metadata-eval 98.1%

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

15. Simplified 98.1%

    \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{1 + x}} \]
16. Add Preprocessing

Alternative 4: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

   1. flip-- 37.8%

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

   2. add-sqr-sqrt 38.0%

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

   3. add-sqr-sqrt 39.2%

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

   4. add-sqr-sqrt 39.2%

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

   5. hypot-1-def 39.2%

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

6. Applied egg-rr 39.2%

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

   1. associate--l+ 81.8%

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

   2. +-inverses 81.8%

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

   3. metadata-eval 81.8%

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

   4. +-commutative 81.8%

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

   5. hypot-undefine 81.8%

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

   6. metadata-eval 81.8%

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

   7. rem-square-sqrt 81.8%

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

8. Simplified 81.8%

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

   1. *-un-lft-identity 81.8%

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

   2. sqrt-prod 99.2%

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

   3. times-frac 99.1%

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

10. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. associate-*l/ 99.2%

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

    3. *-lft-identity 99.2%

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

12. Simplified 99.2%

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

13. Taylor expanded in x around inf 96.8%

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

Alternative 5: 36.3% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

5. Taylor expanded in x around inf 79.0%

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

   1. *-commutative 79.0%

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

   2. unpow1/2 79.0%

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

   3. rem-exp-log 76.2%

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

   4. exp-neg 76.2%

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

   5. exp-prod 76.2%

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

   6. distribute-lft-neg-out 76.2%

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

   7. distribute-rgt-neg-in 76.2%

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

   8. metadata-eval 76.2%

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

   9. exp-to-pow 79.0%

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

7. Simplified 79.0%

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

8. Taylor expanded in x around 0 7.9%

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

   1. expm1-log1p-u 7.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{x}\right)\right)} \]

   2. log1p-define 33.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{0.5}{x}\right)}\right) \]

   3. expm1-undefine 33.9%

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

   4. add-exp-log 33.9%

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

10. Applied egg-rr 33.9%

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

    1. associate--l+ 33.9%

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

12. Simplified 33.9%

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

13. Final simplification 33.9%

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

Alternative 6: 7.9% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. frac-sub 37.3%

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

   2. *-un-lft-identity 37.3%

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

   3. *-rgt-identity 37.3%

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

   4. +-commutative 37.3%

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

   5. sqrt-unprod 37.3%

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

   6. +-commutative 37.3%

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

4. Applied egg-rr 37.3%

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

5. Taylor expanded in x around inf 79.0%

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

   1. *-commutative 79.0%

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

   2. unpow1/2 79.0%

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

   3. rem-exp-log 76.2%

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

   4. exp-neg 76.2%

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

   5. exp-prod 76.2%

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

   6. distribute-lft-neg-out 76.2%

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

   7. distribute-rgt-neg-in 76.2%

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

   8. metadata-eval 76.2%

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

   9. exp-to-pow 79.0%

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

7. Simplified 79.0%

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

8. Taylor expanded in x around 0 7.9%

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

Developer Target 1: 98.3% 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 2024158 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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