2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.2%

Time: 7.5s

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: 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: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return (-(-1.0) / t_0) / ((sqrt(x) + t_0) * sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (-(-1.0d0) / t_0) / ((sqrt(x) + t_0) * sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 36.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 5.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (pow x -1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

6. Final simplification 5.7%

   \[\leadsto \sqrt{{x}^{-1}} \]
7. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16} \cdot 1}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{16}}}{{x}^{2}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16}}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   10. lower-/.f64 98.8

       \[\leadsto \frac{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 98.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.3%

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

5. Taylor expanded in x around inf

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

   1. sub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}}{x}}{\sqrt{x + 1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}}\right)\right)}{x}}{\sqrt{x + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{8}}}{x}\right)\right)}{x}}{\sqrt{x + 1}} \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8}}{\mathsf{neg}\left(x\right)}}}{x}}{\sqrt{x + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8} \cdot 1}}{\mathsf{neg}\left(x\right)}}{x}}{\sqrt{x + 1}} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{-1 \cdot x}}}{x}}{\sqrt{x + 1}} \]

   7. rem-square-sqrt N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot x}}{x}}{\sqrt{x + 1}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot x}}{x}}{\sqrt{x + 1}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}}{x}}{\sqrt{x + 1}} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\frac{1}{8} \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}}{x}}{\sqrt{x + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x}}}{\sqrt{x + 1}} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}}{x}}{\sqrt{x + 1}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x}}{\sqrt{x + 1}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}}{x}}{\sqrt{x + 1}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot x}}{x}}{\sqrt{x + 1}} \]

   16. rem-square-sqrt N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{-1} \cdot x}}{x}}{\sqrt{x + 1}} \]

   17. mul-1-neg N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{\mathsf{neg}\left(x\right)}}}{x}}{\sqrt{x + 1}} \]

   18. distribute-neg-frac2 N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{8}}{x}\right)\right)}}{x}}{\sqrt{x + 1}} \]

   19. sub-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   20. lower--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   21. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 5: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (-1.0 / sqrt((x + 1.0))) / fma(-2.0, x, -0.5);
}

Julia

function code(x)
	return Float64(Float64(-1.0 / sqrt(Float64(x + 1.0))) / fma(-2.0, x, -0.5))
end

Wolfram

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

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 36.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-rgt-identity N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

    \[\leadsto \frac{\frac{-1}{\sqrt{x + 1}}}{\mathsf{fma}\left(-2, x, -0.5\right)} \]
11. Add Preprocessing

Alternative 6: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.3%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 7: 81.2% 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 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 81.5%

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

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 81.1%

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

Alternative 8: 36.8% 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 34.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

7. Applied rewrites 33.2%

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

Developer Target 1: 38.5% 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 2024309 
(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)))))