2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.4%

Time: 11.5s

Alternatives: 10

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: 38.7% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return (1.0 / (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((x + 1.0d0))
    code = (1.0d0 / (x + (t_0 * sqrt(x)))) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

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

Alternative 2: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. distribute-rgt-neg-in 0.0%

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

    3. +-commutative 0.0%

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

    4. unpow2 0.0%

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

    5. rem-square-sqrt 0.0%

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

    6. mul-1-neg 0.0%

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

    7. unsub-neg 0.0%

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

17. Simplified 98.8%

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

18. Final simplification 98.8%

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

Alternative 3: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-0.125 + \frac{0.0625}{x}}{x} - -0.5}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (/ (+ -0.125 (/ 0.0625 x)) x) -0.5) x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return ((((-0.125 + (0.0625 / x)) / x) - -0.5) / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((((-0.125d0) + (0.0625d0 / x)) / x) - (-0.5d0)) / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((((-0.125 + (0.0625 / x)) / x) - -0.5) / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((((-0.125 + (0.0625 / x)) / x) - -0.5) / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(-0.125 + Float64(0.0625 / x)) / x) - -0.5) / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((((-0.125 + (0.0625 / x)) / x) - -0.5) / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(-0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -0.5), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around -inf 0.0%

    \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2} \cdot {\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{2}} + 0.25 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{3}}}{x} - 0.5 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{2}}}{x} + \frac{1}{{\left(\sqrt{-1}\right)}^{2} - 1}}{x}}}{\sqrt{1 + x}} \]

17. Simplified 98.8%

    \[\leadsto \frac{\color{blue}{\frac{-0.5 - \frac{\frac{0.0625}{x} + -0.125}{x}}{-x}}}{\sqrt{1 + x}} \]

18. Final simplification 98.8%

    \[\leadsto \frac{\frac{\frac{-0.125 + \frac{0.0625}{x}}{x} - -0.5}{x}}{\sqrt{x + 1}} \]
19. Add Preprocessing

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around inf 98.3%

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

    1. associate-*r/ 98.3%

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

    2. metadata-eval 98.3%

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

17. Simplified 98.3%

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

18. Final simplification 98.3%

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

Alternative 5: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x \cdot \left(\frac{0.5}{x} + 2\right)}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (* x (+ (/ 0.5 x) 2.0))) (sqrt (+ x 1.0))))

C

double code(double x) {
	return (1.0 / (x * ((0.5 / x) + 2.0))) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x * ((0.5d0 / x) + 2.0d0))) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return (1.0 / (x * ((0.5 / x) + 2.0))) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return (1.0 / (x * ((0.5 / x) + 2.0))) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x * Float64(Float64(0.5 / x) + 2.0))) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x * ((0.5 / x) + 2.0))) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x * N[(N[(0.5 / x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around inf 98.2%

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

    1. associate-*r/ 98.2%

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

    2. metadata-eval 98.2%

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

17. Simplified 98.2%

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

18. Final simplification 98.2%

    \[\leadsto \frac{\frac{1}{x \cdot \left(\frac{0.5}{x} + 2\right)}}{\sqrt{x + 1}} \]
19. Add Preprocessing

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around inf 98.2%

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

    1. associate-*r/ 98.2%

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

    2. metadata-eval 98.2%

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

17. Simplified 98.2%

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

18. Final simplification 98.2%

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

Alternative 7: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. *-un-lft-identity 40.5%

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

   3. +-commutative 40.5%

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

   4. *-rgt-identity 40.5%

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

   5. sqrt-unprod 40.5%

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

   6. +-commutative 40.5%

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

4. Applied egg-rr 40.5%

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

   1. add-cube-cbrt 40.5%

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

   2. sqrt-prod 40.5%

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

   3. times-frac 40.5%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{1 + x} - \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x}} \cdot \frac{\sqrt[3]{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{1 + x}}} \]

   4. pow2 40.5%

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

6. Applied egg-rr 40.5%

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

   1. associate-*l/ 40.5%

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

   2. associate-*r/ 40.5%

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

   3. unpow2 40.5%

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

   4. rem-3cbrt-lft 40.5%

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

   5. div-sub 40.5%

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

   6. *-inverses 40.5%

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

8. Simplified 40.5%

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

9. Taylor expanded in x around inf 98.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - 0.375 \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
10. Step-by-step derivation

    1. associate-*r/ 98.0%

       \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.375 \cdot 1}{x}}}{x}}{\sqrt{x}} \]

    2. metadata-eval 98.0%

       \[\leadsto \frac{\frac{0.5 - \frac{\color{blue}{0.375}}{x}}{x}}{\sqrt{x}} \]

11. Simplified 98.0%

    \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 8: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.5e+122) (pow x -0.5) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = x ** (-0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.pow(x, -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Power[x, -0.5], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 17.4%

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

   3. Taylor expanded in x around 0 7.8%

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

      1. inv-pow 7.8%

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

      2. sqrt-pow1 7.8%

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

      3. metadata-eval 7.8%

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

      4. *-un-lft-identity 7.8%

         \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]

   5. Applied egg-rr 7.8%

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

      1. *-lft-identity 7.8%

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

   7. Simplified 7.8%

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

   if 8.50000000000000003e122 < x

   1. Initial program 56.9%

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

      1. inv-pow 56.9%

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

      2. add-cube-cbrt 21.6%

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

      3. unpow-prod-down 15.3%

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

      4. fma-neg 4.3%

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

      5. cbrt-prod 4.2%

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

      6. add-sqr-sqrt 4.3%

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

      7. distribute-neg-frac 4.3%

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

      8. metadata-eval 4.3%

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

      9. +-commutative 4.3%

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

   4. Applied egg-rr 4.3%

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

   5. Taylor expanded in x around inf 56.9%

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

      1. distribute-rgt1-in 56.9%

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

      2. metadata-eval 56.9%

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

      3. mul0-lft 56.9%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 56.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. frac-sub 40.5%

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

   2. clear-num 40.5%

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

   3. sqrt-unprod 40.5%

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

   4. +-commutative 40.5%

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

   5. *-un-lft-identity 40.5%

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

   6. +-commutative 40.5%

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

   7. *-rgt-identity 40.5%

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

4. Applied egg-rr 40.5%

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

   1. associate-/r/ 40.5%

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

6. Simplified 40.5%

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

   1. flip-- 41.7%

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

   2. add-sqr-sqrt 42.4%

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

   3. add-sqr-sqrt 44.1%

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

   4. add-sqr-sqrt 44.1%

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

   5. hypot-1-def 44.1%

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

8. Applied egg-rr 44.1%

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

   1. associate--l+ 85.2%

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

   2. +-inverses 85.2%

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

   3. metadata-eval 85.2%

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

   4. +-commutative 85.2%

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

   5. hypot-undefine 85.2%

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

   6. metadata-eval 85.2%

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

   7. rem-square-sqrt 85.2%

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

10. Simplified 85.2%

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

    1. associate-*l/ 85.3%

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

    2. *-un-lft-identity 85.3%

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

    3. sqrt-prod 99.3%

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

    4. associate-/r* 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-/l/ 99.2%

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

    2. distribute-rgt-in 99.2%

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

    3. rem-square-sqrt 99.4%

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

14. Simplified 99.4%

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

15. Taylor expanded in x around inf 95.9%

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

16. Final simplification 95.9%

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

Alternative 10: 35.7% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 40.4%

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

   1. inv-pow 40.4%

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

   2. add-cube-cbrt 19.4%

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

   3. unpow-prod-down 15.8%

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

   4. fma-neg 9.4%

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

   5. cbrt-prod 9.6%

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

   6. add-sqr-sqrt 9.4%

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

   7. distribute-neg-frac 9.4%

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

   8. metadata-eval 9.4%

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

   9. +-commutative 9.4%

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

4. Applied egg-rr 9.4%

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

5. Taylor expanded in x around inf 34.8%

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

   1. distribute-rgt1-in 34.8%

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

   2. metadata-eval 34.8%

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

   3. mul0-lft 34.8%

      \[\leadsto \color{blue}{0} \]

7. Simplified 34.8%

   \[\leadsto \color{blue}{0} \]
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 2024106 
(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)))))