2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.5%

Time: 12.3s

Alternatives: 4

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: 38.5% 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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. frac-sub 33.4%

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

   2. *-rgt-identity 33.4%

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

   3. *-un-lft-identity 33.4%

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

   4. +-commutative 33.4%

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

   5. sqrt-unprod 33.4%

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

   6. +-commutative 33.4%

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

4. Applied egg-rr 33.4%

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

   1. flip-- 33.8%

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

   2. add-sqr-sqrt 34.8%

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

   3. add-sqr-sqrt 35.0%

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

   4. add-sqr-sqrt 35.0%

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

   5. hypot-1-def 35.0%

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

6. Applied egg-rr 35.0%

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

   1. associate--l+ 83.5%

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

   2. +-inverses 83.5%

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

   3. metadata-eval 83.5%

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

   4. +-commutative 83.5%

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

   5. hypot-undefine 83.5%

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

   6. metadata-eval 83.5%

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

   7. rem-square-sqrt 83.5%

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

8. Simplified 83.5%

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

   1. clear-num 83.5%

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

   2. associate-/r/ 83.4%

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

   3. distribute-rgt-in 83.4%

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

   4. *-un-lft-identity 83.4%

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

   5. add-sqr-sqrt 83.4%

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

   6. hypot-define 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x * (1.0 + (0.5 / x)));
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[(1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

   1. frac-sub 33.4%

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

   2. *-rgt-identity 33.4%

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

   3. *-un-lft-identity 33.4%

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

   4. +-commutative 33.4%

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

   5. sqrt-unprod 33.4%

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

   6. +-commutative 33.4%

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

4. Applied egg-rr 33.4%

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

   1. flip-- 33.8%

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

   2. add-sqr-sqrt 34.8%

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

   3. add-sqr-sqrt 35.0%

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

   4. add-sqr-sqrt 35.0%

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

   5. hypot-1-def 35.0%

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

6. Applied egg-rr 35.0%

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

   1. associate--l+ 83.5%

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

   2. +-inverses 83.5%

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

   3. metadata-eval 83.5%

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

   4. +-commutative 83.5%

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

   5. hypot-undefine 83.5%

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

   6. metadata-eval 83.5%

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

   7. rem-square-sqrt 83.5%

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

8. Simplified 83.5%

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

9. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. frac-sub 33.4%

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

   2. *-rgt-identity 33.4%

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

   3. *-un-lft-identity 33.4%

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

   4. +-commutative 33.4%

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

   5. sqrt-unprod 33.4%

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

   6. +-commutative 33.4%

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

4. Applied egg-rr 33.4%

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

   1. flip-- 33.8%

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

   2. add-sqr-sqrt 34.8%

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

   3. add-sqr-sqrt 35.0%

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

   4. add-sqr-sqrt 35.0%

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

   5. hypot-1-def 35.0%

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

6. Applied egg-rr 35.0%

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

   1. associate--l+ 83.5%

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

   2. +-inverses 83.5%

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

   3. metadata-eval 83.5%

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

   4. +-commutative 83.5%

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

   5. hypot-undefine 83.5%

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

   6. metadata-eval 83.5%

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

   7. rem-square-sqrt 83.5%

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

8. Simplified 83.5%

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

9. Taylor expanded in x around inf 98.8%

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

10. Taylor expanded in x around inf 98.8%

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

11. Final simplification 98.8%

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

Alternative 4: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. expm1-log1p-u 33.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-undefine 4.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. inv-pow 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. sqrt-pow2 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. sub-neg 4.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} + \left(-1\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. log1p-undefine 4.2%

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

   3. rem-exp-log 4.2%

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

   4. +-commutative 4.2%

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

   5. metadata-eval 4.2%

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

   6. associate-+l+ 24.7%

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

   7. metadata-eval 24.7%

      \[\leadsto \left({x}^{-0.5} + \color{blue}{0}\right) - \frac{1}{\sqrt{x + 1}} \]

   8. +-rgt-identity 24.7%

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

6. Simplified 24.7%

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

7. Taylor expanded in x around 0 5.6%

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

   1. rem-exp-log 5.6%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\log x}}}} \]

   2. exp-neg 5.6%

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

   3. unpow1/2 5.6%

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

   4. exp-prod 5.6%

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

   5. distribute-lft-neg-out 5.6%

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

   6. distribute-rgt-neg-in 5.6%

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

   7. metadata-eval 5.6%

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

   8. exp-to-pow 5.6%

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

9. Simplified 5.6%

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

10. Final simplification 5.6%

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

Developer target: 98.2% 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 2024081 
(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)))))