2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.6%

Time: 9.0s

Alternatives: 7

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

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

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 (sqrt (/ 1.0 x))) x)
     (/ (- (+ x 1.0) x) (* (+ t_0 (sqrt x)) (sqrt (fma x x x)))))))

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 * sqrt((1.0 / x))) / x;
	} else {
		tmp = ((x + 1.0) - x) / ((t_0 + sqrt(x)) * sqrt(fma(x, x, x)));
	}
	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 * sqrt(Float64(1.0 / x))) / x);
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64(Float64(t_0 + sqrt(x)) * sqrt(fma(x, x, x))));
	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 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $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 34.5%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 99.6%

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

   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%

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

   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 60.4%

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

   4. Applied rewrites 99.4%

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

4. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

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 (sqrt (/ 1.0 x))) x)
     (/ (/ (- (+ x 1.0) x) (+ (sqrt (fma x x x)) x)) 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 * sqrt((1.0 / x))) / x;
	} else {
		tmp = (((x + 1.0) - x) / (sqrt(fma(x, x, x)) + x)) / 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 * sqrt(Float64(1.0 / x))) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) - x) / Float64(sqrt(fma(x, x, x)) + x)) / 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 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 34.5%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 99.6%

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

   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%

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

   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 60.4%

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

   4. Applied rewrites 99.2%

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

4. Final simplification 99.7%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-sqrt.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 97.9%

    \[\leadsto \frac{\frac{\frac{-0.5}{x} - -0.5}{\sqrt{x}}}{x} \]
13. Add Preprocessing

Alternative 4: 97.5% accurate, 1.3× speedup

Math

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

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 35.7%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-sqrt.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

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 97.9%

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

11. Final simplification 97.9%

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

Alternative 5: 97.4% 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 35.7%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-sqrt.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 97.8%

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

Alternative 6: 81.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.1%

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

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 80.9%

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

9. Final simplification 80.9%

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

Alternative 7: 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 35.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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: 98.5% 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: 38.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
end

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(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)))))