2isqrt (example 3.6)

Percentage Accurate: 39.0% → 98.8%

Time: 7.8s

Alternatives: 8

Speedup: 1.5×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 39.0% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(0.5 + \frac{0.3125}{x \cdot x}\right) - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (+ 0.5 (/ 0.3125 (* x x))) (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return (((0.5 + (0.3125 / (x * x))) - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 + (0.3125d0 / (x * x))) - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return (((0.5 + (0.3125 / (x * x))) - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return (((0.5 + (0.3125 / (x * x))) - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.3125 / Float64(x * x))) - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 + (0.3125 / (x * x))) - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.3125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. *-inverses N/A

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

   5. lower--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. sqrt-undiv N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 38.3

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

6. Applied rewrites 38.3%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]

9. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.3125}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]

10. Final simplification 99.2%

    \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{x \cdot x}\right) - \frac{0.375}{x}}{x}}{\sqrt{x}} \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t\_0} \leq 0:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(2, x, 0.5\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (/ (/ 0.5 x) (sqrt x))
     (/ (- (+ x 1.0) x) (* (fma 2.0 x 0.5) t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = (0.5 / x) / sqrt(x);
	} else {
		tmp = ((x + 1.0) - x) / (fma(2.0, x, 0.5) * t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64(fma(2.0, x, 0.5) * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(2.0 * x + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 36.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 58.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. rem-square-sqrt N/A

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

      7. unpow2 N/A

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

      8. sub-neg N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lft-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

   9. Applied rewrites 85.3%

      \[\leadsto \frac{1}{\frac{\sqrt{x + 1}}{\frac{\left(x + 1\right) - x}{\color{blue}{\mathsf{fma}\left(2, x, 0.5\right)}}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. clear-num N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lift-+.f64 N/A

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

       10. lower-*.f64 85.4

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

       11. lift-+.f64 N/A

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

       12. +-commutative N/A

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

       13. lift-+.f64 85.4

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

   11. Applied rewrites 85.4%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} \cdot \mathsf{fma}\left(2, x, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(2, x, 0.5\right) \cdot \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{x}{0.5 - \frac{0.375}{x}}}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (/ x (- 0.5 (/ 0.375 x)))) (sqrt x)))

C

double code(double x) {
	return (1.0 / (x / (0.5 - (0.375 / x)))) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x / (0.5d0 - (0.375d0 / x)))) / sqrt(x)
end function

Java

public static double code(double x) {
	return (1.0 / (x / (0.5 - (0.375 / x)))) / Math.sqrt(x);
}

Python

def code(x):
	return (1.0 / (x / (0.5 - (0.375 / x)))) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x / Float64(0.5 - Float64(0.375 / x)))) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x / (0.5 - (0.375 / x)))) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x / N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}\right)\right)}}{x}}{\sqrt{x}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{\color{blue}{x \cdot x}}}{x}}{\sqrt{x}} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{x}}{x}}}{x}}{\sqrt{x}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{\color{blue}{x \cdot \left(\frac{1}{8} + \frac{1}{4}\right)}}{x}}{x}}{x}}{\sqrt{x}} \]

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. times-frac N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 98.3

       \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

7. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
8. Step-by-step derivation

9. Applied rewrites 98.3%

   \[\leadsto \frac{\frac{1}{\color{blue}{\frac{x}{0.5 - \frac{0.375}{x}}}}}{\sqrt{x}} \]
10. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}\right)\right)}}{x}}{\sqrt{x}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{\color{blue}{x \cdot x}}}{x}}{\sqrt{x}} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{x}}{x}}}{x}}{\sqrt{x}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{\color{blue}{x \cdot \left(\frac{1}{8} + \frac{1}{4}\right)}}{x}}{x}}{x}}{\sqrt{x}} \]

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. times-frac N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 98.3

       \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

7. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
8. Add Preprocessing

Alternative 5: 97.4% 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 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.6

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

7. Applied rewrites 96.6%

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

Alternative 6: 82.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 84.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 82.4%

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

9. Final simplification 82.4%

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

Alternative 7: 37.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.9

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

5. Applied rewrites 5.9%

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

7. Applied rewrites 35.4%

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

Alternative 8: 5.7% 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 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.9

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

5. Applied rewrites 5.9%

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

Developer Target 1: 39.1% 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 2024308 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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