2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.6%

Time: 8.9s

Alternatives: 8

Speedup: 1.5×

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: 37.8% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 34.0%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. lift-+.f64 N/A

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

    6. lift-+.f64 N/A

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

    7. +-commutative N/A

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

    8. lift-+.f64 N/A

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

    9. +-commutative N/A

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

    10. lift-+.f64 N/A

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

    11. distribute-lft-in N/A

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

    12. lift-+.f64 N/A

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

    13. lift-sqrt.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. lift-sqrt.f64 N/A

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

    16. rem-square-sqrt N/A

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

    17. lower-fma.f64 N/A

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

    18. lift-+.f64 N/A

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

    19. +-commutative N/A

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

    20. lift-+.f64 N/A

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

    21. +-commutative N/A

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

    22. lift-+.f64 99.5

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

11. Applied rewrites 99.5%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

4. Applied rewrites 34.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{\sqrt{x} + \sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{\sqrt{x} + \sqrt{x + 1}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{1 - \frac{1}{2} \cdot \frac{1}{x}}}{x}}{\sqrt{x} + \sqrt{x + 1}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1 - \color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}}{x}}{\sqrt{x} + \sqrt{x + 1}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1 - \frac{\color{blue}{\frac{1}{2}}}{x}}{x}}{\sqrt{x} + \sqrt{x + 1}} \]

   5. lower-/.f64 98.3

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (* 0.5 (sqrt x)) (* x x))
   (/ (/ 1.0 x) (+ 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (0.5 * sqrt(x)) / (x * x);
	} else {
		tmp = (1.0 / x) / (1.0 + sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = (0.5d0 * sqrt(x)) / (x * x)
    else
        tmp = (1.0d0 / x) / (1.0d0 + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (0.5 * Math.sqrt(x)) / (x * x);
	} else {
		tmp = (1.0 / x) / (1.0 + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = (0.5 * math.sqrt(x)) / (x * x)
	else:
		tmp = (1.0 / x) / (1.0 + math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(0.5 * sqrt(x)) / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(1.0 + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (0.5 * sqrt(x)) / (x * x);
	else
		tmp = (1.0 / x) / (1.0 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 58.0%

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 62.3

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

   10. Applied rewrites 62.3%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\sqrt{x} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{\frac{1}{x}}}{-2 \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 34.0%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 97.2

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

10. Applied rewrites 97.2%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{-2 \cdot x}} \]
11. Add Preprocessing

Alternative 5: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{2 \cdot \sqrt{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

4. Applied rewrites 34.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.2

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \sqrt{x}}} \]

   2. lower-sqrt.f64 97.1

      \[\leadsto \frac{\frac{1}{x}}{2 \cdot \color{blue}{\sqrt{x}}} \]

10. Applied rewrites 97.1%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \sqrt{x}}} \]
11. Add Preprocessing

Alternative 6: 80.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 79.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 78.3%

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

Alternative 7: 36.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.8

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

5. Applied rewrites 5.8%

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

7. Applied rewrites 29.9%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
8. Add Preprocessing

Alternative 8: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.8

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

5. Applied rewrites 5.8%

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

Developer Target 1: 37.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024278 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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