2isqrt (example 3.6)

Percentage Accurate: 40.1% → 99.6%

Time: 11.7s

Alternatives: 5

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: 40.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / hypot(sqrt(x), x)) / (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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. frac-sub 37.2%

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

   2. *-un-lft-identity 37.2%

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

   3. *-rgt-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. sqrt-unprod 37.2%

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

   6. +-commutative 37.2%

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

4. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}}} \]
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 38.3%

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

   3. add-sqr-sqrt 38.9%

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

6. Applied egg-rr 38.9%

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

   1. associate--l+ 84.2%

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

   2. +-inverses 84.2%

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

   3. metadata-eval 84.2%

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

   4. +-commutative 84.2%

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

8. Simplified 84.2%

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

   1. *-un-lft-identity 84.2%

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

   2. associate-/l/ 84.2%

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

   3. *-commutative 84.2%

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

   4. distribute-rgt-in 84.2%

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

   5. *-un-lft-identity 84.2%

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

   6. add-sqr-sqrt 84.2%

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

   7. hypot-define 98.7%

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

10. Applied egg-rr 98.7%

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

    1. *-lft-identity 98.7%

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

    2. associate-/r* 99.6%

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

    3. *-lft-identity 99.6%

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

    4. associate-*l/ 99.6%

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

    5. associate-*r/ 99.7%

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

    6. *-rgt-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. frac-sub 37.2%

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

   2. *-un-lft-identity 37.2%

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

   3. *-rgt-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. sqrt-unprod 37.2%

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

   6. +-commutative 37.2%

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

4. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}}} \]
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 38.3%

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

   3. add-sqr-sqrt 38.9%

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

6. Applied egg-rr 38.9%

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

   1. associate--l+ 84.2%

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

   2. +-inverses 84.2%

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

   3. metadata-eval 84.2%

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

   4. +-commutative 84.2%

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

8. Simplified 84.2%

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

   1. *-un-lft-identity 84.2%

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

   2. associate-/l/ 84.2%

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

   3. *-commutative 84.2%

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

   4. distribute-rgt-in 84.2%

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

   5. *-un-lft-identity 84.2%

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

   6. add-sqr-sqrt 84.2%

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

   7. hypot-define 98.7%

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

10. Applied egg-rr 98.7%

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

    1. *-lft-identity 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. frac-sub 37.2%

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

   2. *-un-lft-identity 37.2%

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

   3. *-rgt-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. sqrt-unprod 37.2%

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

   6. +-commutative 37.2%

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

4. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}}} \]
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 38.3%

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

   3. add-sqr-sqrt 38.9%

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

6. Applied egg-rr 38.9%

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

   1. associate--l+ 84.2%

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

   2. +-inverses 84.2%

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

   3. metadata-eval 84.2%

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

   4. +-commutative 84.2%

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

8. Simplified 84.2%

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

   1. *-un-lft-identity 84.2%

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

   2. associate-/l/ 84.2%

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

   3. *-commutative 84.2%

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

   4. distribute-rgt-in 84.2%

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

   5. *-un-lft-identity 84.2%

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

   6. add-sqr-sqrt 84.2%

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

   7. hypot-define 98.7%

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

10. Applied egg-rr 98.7%

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

    1. *-lft-identity 98.7%

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

    2. associate-/r* 99.6%

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

    3. *-lft-identity 99.6%

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

    4. associate-*l/ 99.6%

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

    5. associate-*r/ 99.7%

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

    6. *-rgt-identity 99.7%

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

12. Simplified 99.7%

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

13. Taylor expanded in x around inf 98.7%

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

    1. associate-*r/ 98.7%

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

    2. metadata-eval 98.7%

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

15. Simplified 98.7%

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

16. Final simplification 98.7%

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

Alternative 4: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

3. Taylor expanded in x around inf 63.8%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 63.8%

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

   2. pow-flip 64.4%

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

   3. sqrt-pow1 98.0%

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

   4. metadata-eval 98.0%

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

   5. metadata-eval 98.0%

      \[\leadsto 0.5 \cdot \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \]

5. Applied egg-rr 98.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \]
6. Step-by-step derivation

   1. *-lft-identity 98.0%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

7. Simplified 98.0%

   \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

8. Final simplification 98.0%

   \[\leadsto 0.5 \cdot {x}^{-1.5} \]
9. Add Preprocessing

Alternative 5: 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 37.2%

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

   1. frac-sub 37.2%

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

   2. *-un-lft-identity 37.2%

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

   3. *-rgt-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. sqrt-unprod 37.2%

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

   6. +-commutative 37.2%

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

4. Applied egg-rr 37.2%

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

5. Taylor expanded in x around inf 82.4%

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

   1. *-commutative 82.4%

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

7. Simplified 82.4%

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

8. Taylor expanded in x around 0 7.9%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]

9. Final simplification 7.9%

   \[\leadsto \frac{0.5}{x} \]
10. 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 2024109 
(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)))))