2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.7%

Time: 11.1s

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: 38.7% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{1}{\sqrt{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, x\right)} + \sqrt{x \cdot \mathsf{fma}\left(x, x, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+90)
   (/ 1.0 (+ (sqrt (* (+ x 1.0) (fma x x x))) (sqrt (* x (fma x x x)))))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+90) {
		tmp = 1.0 / (sqrt(((x + 1.0) * fma(x, x, x))) + sqrt((x * fma(x, x, x))));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+90)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(x + 1.0) * fma(x, x, x))) + sqrt(Float64(x * 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, 1e+90], N[(1.0 / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] * N[(x * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(x * 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 < 9.99999999999999966e89

   1. Initial program 12.2%

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

   4. Applied egg-rr 17.3%

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

      1. associate--l+ N/A

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

      2. +-inverses N/A

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

      3. metadata-eval 99.3

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

   6. Applied egg-rr 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

   8. Applied egg-rr 99.5%

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

   if 9.99999999999999966e89 < x

   1. Initial program 50.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 99.8

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

   11. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{1}{\sqrt{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, x\right)} + \sqrt{x \cdot \mathsf{fma}\left(x, x, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.7e+149)
   (/ 1.0 (* (sqrt (fma x x x)) (+ (sqrt (+ x 1.0)) (sqrt x))))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.7e+149) {
		tmp = 1.0 / (sqrt(fma(x, x, x)) * (sqrt((x + 1.0)) + sqrt(x)));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.7e+149)
		tmp = Float64(1.0 / Float64(sqrt(fma(x, x, x)) * Float64(sqrt(Float64(x + 1.0)) + sqrt(x))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.7e+149], N[(1.0 / N[(N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $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 < 3.69999999999999978e149

   1. Initial program 9.3%

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

   4. Applied egg-rr 12.3%

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

      1. associate--l+ N/A

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

      2. +-inverses N/A

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

      3. metadata-eval 99.4

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

   6. Applied egg-rr 99.4%

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

   if 3.69999999999999978e149 < x

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 99.9

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

   11. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

4. Applied egg-rr 41.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. associate--l+ N/A

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

   14. +-inverses N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 84.0

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 84.0

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

6. Applied egg-rr 84.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 98.0

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 4: 97.6% 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 39.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 82.4

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

5. Simplified 82.4%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

8. Simplified 98.1%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-/.f64 98.0

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

11. Simplified 98.0%

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

Alternative 5: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 82.4

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

5. Simplified 82.4%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

8. Simplified 98.1%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-/.f64 98.0

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

11. Simplified 98.0%

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

    1. lift-/.f64 N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-/.f64 98.0

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

    5. lift-*.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. sqrt-div N/A

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

    9. metadata-eval N/A

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

    10. lift-sqrt.f64 N/A

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

    11. un-div-inv N/A

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

    12. lower-/.f64 98.0

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

13. Applied egg-rr 98.0%

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

Alternative 6: 37.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.8e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.8e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.8d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.8e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.8e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.8e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.8e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.8e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999985e153

   1. Initial program 9.2%

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

   4. Applied egg-rr 12.1%

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

      1. associate--l+ N/A

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

      2. +-inverses N/A

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

      3. metadata-eval 99.4

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot \left(1 + \sqrt{x}\right)}} \]
   8. 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 8.3

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

   9. Simplified 8.3%

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

   if 4.79999999999999985e153 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 53.1

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

   5. Simplified 53.1%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-inverses 68.7

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

   7. Applied egg-rr 68.7%

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

Alternative 7: 96.4% 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 39.9%

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

4. Applied egg-rr 41.3%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 39.7

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

7. Simplified 39.7%

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

   1. associate--l+ N/A

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

   2. +-inverses N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 65.1

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

   11. lift-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. cube-unmult N/A

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

   15. sqrt-pow1 N/A

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

   16. sqrt-pow2 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. unpow3 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. lift-sqrt.f64 N/A

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

   21. rem-square-sqrt N/A

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

   22. lower-*.f64 97.1

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

9. Applied egg-rr 97.1%

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

Alternative 8: 35.8% accurate, 49.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 29.6

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

5. Simplified 29.6%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-inverses 37.6

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

7. Applied egg-rr 37.6%

   \[\leadsto \color{blue}{0} \]
8. 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: 38.8% 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 2024208 
(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)))))