2isqrt (example 3.6)

Percentage Accurate: 38.2% → 99.4%

Time: 10.8s

Alternatives: 11

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: 38.2% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 38.1%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

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

   13. metadata-eval N/A

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

   14. associate-/r* N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 84.9

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

6. Applied rewrites 84.9%

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

   1. distribute-lft1-in N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 38.1%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 99.1

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

7. Applied rewrites 99.1%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 38.1%

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

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. lower-+.f64 37.6

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

7. Applied rewrites 37.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. lift-+.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-/l/ N/A

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

   11. metadata-eval N/A

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

   12. +-inverses N/A

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

   13. associate--l+ N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. associate-/l/ N/A

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

   17. lift-/.f64 N/A

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

   18. div-inv N/A

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

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

Alternative 4: 97.8% 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 36.3%

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

4. Applied rewrites 38.1%

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

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. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 5: 97.8% 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 36.3%

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

4. Applied rewrites 38.1%

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

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. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

Alternative 6: 97.7% 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 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 84.4%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 83.9

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

8. Applied rewrites 83.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. sqrt-pow1 N/A

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

   13. cube-unmult N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. un-div-inv N/A

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

   17. lift-*.f64 N/A

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

   18. sqrt-prod N/A

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

   19. lift-sqrt.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. sqrt-prod N/A

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

   22. rem-square-sqrt N/A

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

10. Applied rewrites 98.6%

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

Alternative 7: 97.7% 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 36.3%

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

4. Applied rewrites 38.1%

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

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. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 98.6

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

10. Applied rewrites 98.6%

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

Alternative 8: 96.6% 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 36.3%

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

4. Applied rewrites 38.1%

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

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. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 97.7

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

9. Applied rewrites 97.7%

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

10. Final simplification 97.7%

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

Alternative 9: 96.5% 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 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 84.4%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 83.9

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

8. Applied rewrites 83.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. sqrt-pow1 N/A

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

   13. cube-unmult N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. un-div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. sqrt-prod N/A

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

   20. lift-sqrt.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. sqrt-prod N/A

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

   23. rem-square-sqrt N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 97.6

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

10. Applied rewrites 97.6%

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

Alternative 10: 37.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.5e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+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 <= 6.5d+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 <= 6.5e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.5e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+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 <= 6.5e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999972e153

   1. Initial program 7.4%

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

   4. Applied rewrites 10.7%

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

   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. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 8.5

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

   10. Applied rewrites 8.5%

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

   if 6.49999999999999972e153 < x

   1. Initial program 68.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. lower-sqrt.f64 N/A

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

      2. lower-/.f64 51.7

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

   5. Applied rewrites 51.7%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
   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. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-inverses 68.7

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

   7. Applied rewrites 68.7%

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

Alternative 11: 35.4% 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 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lower-sqrt.f64 N/A

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

   2. lower-/.f64 26.7

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

5. Applied rewrites 26.7%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
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. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-inverses 34.7

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

7. Applied rewrites 34.7%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.2% 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 2024214 
(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)))))