2isqrt (example 3.6)

Percentage Accurate: 39.3% → 99.7%

Time: 15.3s

Alternatives: 7

Speedup: 2.0×

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: 39.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5e+23)
   (/ (/ 1.0 (* x (+ 1.0 x))) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 4.5e+23) {
		tmp = (1.0 / (x * (1.0 + x))) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.5d+23) then
        tmp = (1.0d0 / (x * (1.0d0 + x))) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.5e+23) {
		tmp = (1.0 / (x * (1.0 + x))) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.5e+23:
		tmp = (1.0 / (x * (1.0 + x))) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5e+23)
		tmp = Float64(Float64(1.0 / Float64(x * Float64(1.0 + x))) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.5e+23)
		tmp = (1.0 / (x * (1.0 + x))) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	else
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.5e+23], N[(N[(1.0 / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $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 < 4.49999999999999979e23

   1. Initial program 40.3%

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

      1. flip-- 40.7%

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

      2. frac-times 41.1%

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

      3. metadata-eval 41.1%

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

      4. add-sqr-sqrt 41.6%

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

      5. frac-times 41.7%

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

      6. metadata-eval 41.7%

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

      7. add-sqr-sqrt 41.7%

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

      8. +-commutative 41.7%

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

      9. inv-pow 41.7%

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

      10. sqrt-pow2 41.7%

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

      11. metadata-eval 41.7%

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

      12. pow1/2 41.7%

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

      13. pow-flip 41.7%

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

      14. +-commutative 41.7%

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

      15. metadata-eval 41.7%

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

   4. Applied egg-rr 41.7%

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

      1. frac-sub 65.2%

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

      2. *-un-lft-identity 65.2%

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in x around 0 99.3%

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

   if 4.49999999999999979e23 < x

   1. Initial program 37.6%

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

   3. Taylor expanded in x around inf 78.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.7%

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

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\frac{x \cdot 0.015625 - x \cdot 0.25}{x \cdot -0.625}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

   7. Taylor expanded in x around inf 99.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around inf 79.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.7%

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

5. Taylor expanded in x around 0 98.7%

   \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\frac{-0.125 \cdot \sqrt{x} + 0.5 \cdot \sqrt{x}}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
6. Step-by-step derivation

   1. distribute-rgt-out 98.7%

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

   2. metadata-eval 98.7%

      \[\leadsto \frac{-1 \cdot \frac{\frac{\sqrt{x} \cdot \color{blue}{0.375}}{x}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

7. Applied egg-rr 98.7%

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

Alternative 3: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1 \cdot \frac{\frac{x \cdot 0.125 + x \cdot -0.001953125}{x \cdot 0.25 + \left(x \cdot 0.015625 - \left(x \cdot 0.5\right) \cdot \left(-0.125 \cdot x\right)\right)}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (*
    -1.0
    (/
     (/
      (+ (* x 0.125) (* x -0.001953125))
      (+ (* x 0.25) (- (* x 0.015625) (* (* x 0.5) (* -0.125 x)))))
     x))
   (* 0.5 (sqrt (/ 1.0 x))))
  x))

C

double code(double x) {
	return ((-1.0 * ((((x * 0.125) + (x * -0.001953125)) / ((x * 0.25) + ((x * 0.015625) - ((x * 0.5) * (-0.125 * x))))) / x)) + (0.5 * sqrt((1.0 / x)))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-1.0d0) * ((((x * 0.125d0) + (x * (-0.001953125d0))) / ((x * 0.25d0) + ((x * 0.015625d0) - ((x * 0.5d0) * ((-0.125d0) * x))))) / x)) + (0.5d0 * sqrt((1.0d0 / x)))) / x
end function

Java

public static double code(double x) {
	return ((-1.0 * ((((x * 0.125) + (x * -0.001953125)) / ((x * 0.25) + ((x * 0.015625) - ((x * 0.5) * (-0.125 * x))))) / x)) + (0.5 * Math.sqrt((1.0 / x)))) / x;
}

Python

def code(x):
	return ((-1.0 * ((((x * 0.125) + (x * -0.001953125)) / ((x * 0.25) + ((x * 0.015625) - ((x * 0.5) * (-0.125 * x))))) / x)) + (0.5 * math.sqrt((1.0 / x)))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(Float64(x * 0.125) + Float64(x * -0.001953125)) / Float64(Float64(x * 0.25) + Float64(Float64(x * 0.015625) - Float64(Float64(x * 0.5) * Float64(-0.125 * x))))) / x)) + Float64(0.5 * sqrt(Float64(1.0 / x)))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-1.0 * ((((x * 0.125) + (x * -0.001953125)) / ((x * 0.25) + ((x * 0.015625) - ((x * 0.5) * (-0.125 * x))))) / x)) + (0.5 * sqrt((1.0 / x)))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-1.0 * N[(N[(N[(N[(x * 0.125), $MachinePrecision] + N[(x * -0.001953125), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.25), $MachinePrecision] + N[(N[(x * 0.015625), $MachinePrecision] - N[(N[(x * 0.5), $MachinePrecision] * N[(-0.125 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around inf 79.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.7%

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

6. Applied egg-rr 98.0%

   \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\frac{x \cdot 0.125 + x \cdot -0.001953125}{x \cdot 0.25 + \left(x \cdot 0.015625 - \left(x \cdot 0.5\right) \cdot \left(-0.125 \cdot x\right)\right)}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
7. Add Preprocessing

Alternative 4: 97.6% accurate, 2.0× 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 37.8%

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

3. Taylor expanded in x around inf 79.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.7%

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

6. Applied egg-rr 96.0%

   \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\frac{x \cdot 0.015625 - x \cdot 0.25}{x \cdot -0.625}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

7. Taylor expanded in x around inf 98.0%

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

Alternative 5: 45.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -1.5))

C

double code(double x) {
	return pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x ** (-1.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.5);
}

Python

def code(x):
	return math.pow(x, -1.5)

Julia

function code(x)
	return x ^ -1.5
end

MATLAB

function tmp = code(x)
	tmp = x ^ -1.5;
end

Wolfram

code[x_] := N[Power[x, -1.5], $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around 0 5.6%

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

   1. pow1/2 5.6%

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

   2. pow-to-exp 5.6%

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

   3. log-rec 5.6%

      \[\leadsto e^{\color{blue}{\left(-\log x\right)} \cdot 0.5} \]

5. Applied egg-rr 5.6%

   \[\leadsto \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]
6. Step-by-step derivation

   1. neg-log 5.6%

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

   2. pow-to-exp 5.6%

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

   3. metadata-eval 5.6%

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

   4. pow-div 5.6%

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

   5. pow1 5.6%

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

   6. pow1/2 5.6%

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

   7. add-exp-log 5.6%

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

   8. neg-log 5.6%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 44.1%

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

   11. sqr-neg 44.1%

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

   12. sqrt-unprod 44.1%

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

   13. add-sqr-sqrt 44.1%

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

   14. add-exp-log 44.1%

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

   15. un-div-inv 44.1%

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

   16. metadata-eval 44.1%

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

   17. sqrt-div 44.1%

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

   18. add-sqr-sqrt 44.1%

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

   19. pow3 44.1%

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

7. Applied egg-rr 44.1%

   \[\leadsto \color{blue}{{x}^{-1.5}} \]
8. Add Preprocessing

Alternative 6: 38.4% accurate, 29.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (- x) (- x))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (-x * -x);
}

Python

def code(x):
	return 1.0 / (-x * -x)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(-x) * Float64(-x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around 0 5.6%

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

   1. pow1/2 5.6%

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

   2. pow-to-exp 5.6%

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

   3. log-rec 5.6%

      \[\leadsto e^{\color{blue}{\left(-\log x\right)} \cdot 0.5} \]

5. Applied egg-rr 5.6%

   \[\leadsto \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]
6. Step-by-step derivation

   1. exp-prod 5.6%

      \[\leadsto \color{blue}{{\left(e^{-\log x}\right)}^{0.5}} \]

7. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{\frac{1}{\left(-x\right) \cdot \left(-x\right)}} \]
8. Add Preprocessing

Alternative 7: 7.9% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around 0 5.6%

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

   1. pow1/2 5.6%

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

   2. pow-to-exp 5.6%

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

   3. log-rec 5.6%

      \[\leadsto e^{\color{blue}{\left(-\log x\right)} \cdot 0.5} \]

5. Applied egg-rr 5.6%

   \[\leadsto \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]
6. Step-by-step derivation

   1. exp-prod 5.6%

      \[\leadsto \color{blue}{{\left(e^{-\log x}\right)}^{0.5}} \]

   2. add-sqr-sqrt 0.0%

      \[\leadsto {\left(e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}\right)}^{0.5} \]

   3. sqrt-unprod 3.9%

      \[\leadsto {\left(e^{\color{blue}{\sqrt{\left(-\log x\right) \cdot \left(-\log x\right)}}}\right)}^{0.5} \]

   4. sqr-neg 3.9%

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

   5. sqrt-unprod 3.9%

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

   6. add-sqr-sqrt 3.9%

      \[\leadsto {\left(e^{\color{blue}{\log x}}\right)}^{0.5} \]

   7. add-exp-log 3.9%

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

   8. pow1/2 3.9%

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

   9. add-exp-log 3.9%

      \[\leadsto \sqrt{\color{blue}{e^{\log x}}} \]

   10. add-sqr-sqrt 3.9%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}} \]

   11. sqrt-unprod 3.9%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\log x \cdot \log x}}}} \]

   12. sqr-neg 3.9%

       \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\log x\right) \cdot \left(-\log x\right)}}}} \]

   13. sqrt-unprod 0.0%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}} \]

   14. add-sqr-sqrt 5.6%

       \[\leadsto \sqrt{e^{\color{blue}{-\log x}}} \]

   15. neg-log 5.6%

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

   16. add-sqr-sqrt 5.6%

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

   17. add-exp-log 5.6%

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

7. Applied egg-rr 7.8%

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

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024050 -o generate:simplify
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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