2isqrt (example 3.6)

Percentage Accurate: 38.3% → 98.8%

Time: 11.8s

Alternatives: 10

Speedup: 1.8×

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: 38.3% 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: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 40.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. +-lowering-+.f64 39.0

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

7. Simplified 39.0%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. +-commutative N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 98.4

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

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 40.3%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5

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

7. Simplified 97.5%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 97.5

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

9. Applied egg-rr 97.5%

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

10. Final simplification 97.5%

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

Alternative 3: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 40.3%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5

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

7. Simplified 97.5%

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

Alternative 4: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 80.5%

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

6. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 79.5

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

8. Simplified 79.5%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. pow1/2 N/A

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

   6. inv-pow N/A

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

   7. pow-prod-up N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. sqrt-pow1 N/A

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

   11. inv-pow N/A

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

   12. sqrt-div N/A

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

   13. metadata-eval N/A

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

   14. un-div-inv N/A

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

   15. /-lowering-/.f64 N/A

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

   16. sqrt-lowering-sqrt.f64 97.4

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

10. Applied egg-rr 97.4%

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

Alternative 5: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 40.3%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5

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

7. Simplified 97.5%

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

8. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 97.3

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

10. Simplified 97.3%

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

Alternative 6: 96.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 40.3%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5

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

7. Simplified 97.5%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 96.1

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

9. Applied egg-rr 96.1%

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

10. Final simplification 96.1%

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

Alternative 7: 96.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 80.5%

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

6. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 79.5

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

8. Simplified 79.5%

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

   1. associate-/l* N/A

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

   2. pow1/2 N/A

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

   3. pow2 N/A

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

   4. pow-div N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {x}^{\color{blue}{\frac{-3}{2}}} \]

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. inv-pow N/A

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

   10. pow-pow N/A

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

   11. pow1/2 N/A

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

   12. sqrt-div N/A

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

   13. metadata-eval N/A

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

   14. cube-div N/A

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

   15. metadata-eval N/A

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

   16. un-div-inv N/A

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

   17. /-lowering-/.f64 N/A

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

   18. unpow3 N/A

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

   19. rem-square-sqrt N/A

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

   20. *-lowering-*.f64 N/A

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

   21. sqrt-lowering-sqrt.f64 95.9

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

10. Applied egg-rr 95.9%

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

Alternative 8: 37.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.4e+153) (/ 1.0 (+ x (+ x 0.5))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 1.0 / (x + (x + 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 1.0d0 / (x + (x + 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 1.0 / (x + (x + 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 1.0 / (x + (x + 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(1.0 / Float64(x + Float64(x + 0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 1.0 / (x + (x + 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(1.0 / N[(x + N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 10.6%

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 11.8

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

   7. Simplified 11.8%

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

      1. associate-/l/ N/A

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

      2. associate--l+ N/A

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

      3. +-inverses N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 96.9

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

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 8.5%

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

   if 6.4000000000000003e153 < x

   1. Initial program 65.0%

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

   3. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 44.2

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

   5. Simplified 44.2%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. +-inverses 65.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.4e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 10.6%

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 95.0

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

   7. Simplified 95.0%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 8.5

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

   10. Simplified 8.5%

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

   if 6.4000000000000003e153 < x

   1. Initial program 65.0%

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

   3. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 44.2

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

   5. Simplified 44.2%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. +-inverses 65.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.3% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 N/A

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

   2. /-lowering-/.f64 24.8

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

5. Simplified 24.8%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 35.4

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 35.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 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]

Developer Target 2: 38.3% 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 2024199 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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

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