2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.7%

Time: 13.0s

Alternatives: 3

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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (/ 0.5 x) (/ 0.375 (pow x 2.0))) (sqrt x)))

C

double code(double x) {
	return ((0.5 / x) - (0.375 / pow(x, 2.0))) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 / x) - (0.375d0 / (x ** 2.0d0))) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 / x) - (0.375 / Math.pow(x, 2.0))) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 / x) - (0.375 / math.pow(x, 2.0))) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / x) - Float64(0.375 / (x ^ 2.0))) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / x) - (0.375 / (x ^ 2.0))) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.2%

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

   1. frac-2neg 43.2%

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

   2. metadata-eval 43.2%

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

   3. frac-2neg 43.2%

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

   4. metadata-eval 43.2%

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

   5. frac-sub 43.3%

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

   6. fma-neg 43.3%

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

   7. +-commutative 43.3%

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

   8. +-commutative 43.3%

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

4. Applied egg-rr 43.3%

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

   1. fma-undefine 43.3%

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

   2. distribute-rgt-neg-in 43.3%

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

   3. metadata-eval 43.3%

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

   4. *-rgt-identity 43.3%

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

   5. unsub-neg 43.3%

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

   6. mul-1-neg 43.3%

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

   7. remove-double-neg 43.3%

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

   8. *-lft-identity 43.3%

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

   9. *-lft-identity 43.3%

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

   10. distribute-rgt-neg-out 43.3%

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

   11. *-commutative 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. remove-double-neg 43.3%

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

   14. associate-/r* 43.3%

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

6. Simplified 43.3%

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

7. Taylor expanded in x around inf 99.0%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.0%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

   2. metadata-eval 99.0%

      \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

   3. associate-*r/ 99.0%

      \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}}{\sqrt{x}} \]

   4. metadata-eval 99.0%

      \[\leadsto \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}}{\sqrt{x}} \]

9. Simplified 99.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}}{\sqrt{x}} \]

10. Final simplification 99.0%

    \[\leadsto \frac{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}{\sqrt{x}} \]
11. Add Preprocessing

Alternative 2: 97.9% 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 43.2%

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

   1. frac-2neg 43.2%

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

   2. metadata-eval 43.2%

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

   3. frac-2neg 43.2%

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

   4. metadata-eval 43.2%

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

   5. frac-sub 43.3%

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

   6. fma-neg 43.3%

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

   7. +-commutative 43.3%

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

   8. +-commutative 43.3%

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

4. Applied egg-rr 43.3%

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

   1. fma-undefine 43.3%

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

   2. distribute-rgt-neg-in 43.3%

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

   3. metadata-eval 43.3%

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

   4. *-rgt-identity 43.3%

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

   5. unsub-neg 43.3%

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

   6. mul-1-neg 43.3%

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

   7. remove-double-neg 43.3%

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

   8. *-lft-identity 43.3%

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

   9. *-lft-identity 43.3%

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

   10. distribute-rgt-neg-out 43.3%

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

   11. *-commutative 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. remove-double-neg 43.3%

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

   14. associate-/r* 43.3%

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

6. Simplified 43.3%

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

7. Taylor expanded in x around inf 42.6%

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

   1. associate-*r/ 42.6%

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

   2. metadata-eval 42.6%

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

9. Simplified 42.6%

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

    1. associate--r- 98.3%

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

    2. metadata-eval 98.3%

       \[\leadsto \frac{\color{blue}{0} + \frac{0.5}{x}}{\sqrt{x}} \]

    3. +-lft-identity 98.3%

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

    4. *-un-lft-identity 98.3%

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

    5. add-sqr-sqrt 98.1%

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

    6. times-frac 98.1%

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

    7. pow1/2 98.1%

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

    8. sqrt-pow1 98.1%

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

    9. metadata-eval 98.1%

       \[\leadsto \frac{1}{{x}^{\color{blue}{0.25}}} \cdot \frac{\frac{0.5}{x}}{\sqrt{\sqrt{x}}} \]

    10. pow1/2 98.1%

        \[\leadsto \frac{1}{{x}^{0.25}} \cdot \frac{\frac{0.5}{x}}{\sqrt{\color{blue}{{x}^{0.5}}}} \]

    11. sqrt-pow1 98.1%

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

    12. metadata-eval 98.1%

        \[\leadsto \frac{1}{{x}^{0.25}} \cdot \frac{\frac{0.5}{x}}{{x}^{\color{blue}{0.25}}} \]

11. Applied egg-rr 98.1%

    \[\leadsto \color{blue}{\frac{1}{{x}^{0.25}} \cdot \frac{\frac{0.5}{x}}{{x}^{0.25}}} \]
12. Step-by-step derivation

    1. associate-*l/ 98.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{0.5}{x}}{{x}^{0.25}}}{{x}^{0.25}}} \]

    2. *-lft-identity 98.0%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.5}{x}}{{x}^{0.25}}}}{{x}^{0.25}} \]

    3. associate-/l/ 98.1%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{{x}^{0.25} \cdot x}}}{{x}^{0.25}} \]

    4. pow-plus 98.2%

       \[\leadsto \frac{\frac{0.5}{\color{blue}{{x}^{\left(0.25 + 1\right)}}}}{{x}^{0.25}} \]

    5. metadata-eval 98.2%

       \[\leadsto \frac{\frac{0.5}{{x}^{\color{blue}{1.25}}}}{{x}^{0.25}} \]

13. Simplified 98.2%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{{x}^{1.25}}}{{x}^{0.25}}} \]
14. Step-by-step derivation

    1. associate-/l/ 97.3%

       \[\leadsto \color{blue}{\frac{0.5}{{x}^{0.25} \cdot {x}^{1.25}}} \]

    2. pow-prod-up 97.4%

       \[\leadsto \frac{0.5}{\color{blue}{{x}^{\left(0.25 + 1.25\right)}}} \]

    3. metadata-eval 97.4%

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

    4. metadata-eval 97.4%

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

    5. pow-div 97.4%

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

    6. pow1 97.4%

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

    7. associate-/l* 98.4%

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

    8. *-commutative 98.4%

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

    9. associate-*l/ 98.4%

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

    10. pow1 98.4%

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

    11. pow-div 98.6%

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

    12. metadata-eval 98.6%

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

15. Applied egg-rr 98.6%

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

16. Final simplification 98.6%

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

Alternative 3: 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 43.2%

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

   1. frac-2neg 43.2%

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

   2. metadata-eval 43.2%

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

   3. frac-2neg 43.2%

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

   4. metadata-eval 43.2%

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

   5. frac-sub 43.3%

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

   6. fma-neg 43.3%

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

   7. +-commutative 43.3%

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

   8. +-commutative 43.3%

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

4. Applied egg-rr 43.3%

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

   1. fma-undefine 43.3%

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

   2. distribute-rgt-neg-in 43.3%

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

   3. metadata-eval 43.3%

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

   4. *-rgt-identity 43.3%

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

   5. unsub-neg 43.3%

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

   6. mul-1-neg 43.3%

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

   7. remove-double-neg 43.3%

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

   8. *-lft-identity 43.3%

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

   9. *-lft-identity 43.3%

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

   10. distribute-rgt-neg-out 43.3%

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

   11. *-commutative 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. remove-double-neg 43.3%

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

   14. associate-/r* 43.3%

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

6. Simplified 43.3%

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

7. Taylor expanded in x around 0 5.5%

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

   1. add0 5.5%

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

   2. inv-pow 5.5%

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

   3. sqrt-pow2 5.5%

      \[\leadsto \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} + 0 \]

   4. metadata-eval 5.5%

      \[\leadsto {x}^{\color{blue}{-0.5}} + 0 \]

9. Applied egg-rr 5.5%

   \[\leadsto \color{blue}{{x}^{-0.5} + 0} \]
10. Step-by-step derivation

    1. add0 5.5%

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

11. Simplified 5.5%

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

12. Final simplification 5.5%

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

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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