2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.7%

Time: 9.3s

Alternatives: 8

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: 38.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} \cdot \left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+21)
   (/ 1.0 (* (sqrt (+ 1.0 x)) (+ x (sqrt (fma x x x)))))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+21) {
		tmp = 1.0 / (sqrt((1.0 + x)) * (x + sqrt(fma(x, x, x))));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+21)
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) * Float64(x + sqrt(fma(x, x, x)))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+21], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * N[(x + N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e21

   1. Initial program 45.9%

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

   4. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 5e21 < x

   1. Initial program 39.3%

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \frac{-1}{8}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}}{x} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.7%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 39.1

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

7. Applied rewrites 39.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}}}{x + \frac{1}{2}} \]

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 39.1

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

7. Applied rewrites 39.1%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.1

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

10. Applied rewrites 98.1%

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

Alternative 4: 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 39.7%

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

4. Applied rewrites 41.7%

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

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 98.0

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

7. Applied rewrites 98.0%

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

Alternative 5: 96.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / ((x + 0.5) * (sqrt(x) * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x + 0.5d0) * (sqrt(x) * 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / ((x + 0.5) * (Math.sqrt(x) * 2.0));
}

Python

def code(x):
	return 1.0 / ((x + 0.5) * (math.sqrt(x) * 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x + 0.5) * Float64(sqrt(x) * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x + 0.5) * (sqrt(x) * 2.0));
end

Wolfram

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

Derivation

1. Initial program 39.7%

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

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 39.1

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

7. Applied rewrites 39.1%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. frac-times N/A

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

9. Applied rewrites 97.4%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 96.5

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

12. Applied rewrites 96.5%

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

13. Final simplification 96.5%

    \[\leadsto \frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{x} \cdot 2\right)} \]
14. Add Preprocessing

Alternative 6: 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 39.7%

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 66.5

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

7. Applied rewrites 66.5%

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

9. Applied rewrites 96.4%

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

Alternative 7: 7.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 39.1

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

7. Applied rewrites 39.1%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. frac-times N/A

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

9. Applied rewrites 97.4%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. rem-square-sqrt N/A

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

    4. *-rgt-identity N/A

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

    5. lower-+.f64 N/A

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

    6. lower-sqrt.f64 7.9

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

12. Applied rewrites 7.9%

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

Alternative 8: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.6

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

5. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
6. 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: 38.9% 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 2024232 
(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)))))