2isqrt (example 3.6)

Percentage Accurate: 38.2% → 99.9%

Time: 13.8s

Alternatives: 3

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: 38.2% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9e+15)
   (/
    (/ (+ x (- -1.0 x)) (* x (- -1.0 x)))
    (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 9e+15) {
		tmp = ((x + (-1.0 - x)) / (x * (-1.0 - x))) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9d+15) then
        tmp = ((x + ((-1.0d0) - x)) / (x * ((-1.0d0) - x))) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9e+15) {
		tmp = ((x + (-1.0 - x)) / (x * (-1.0 - x))) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9e+15:
		tmp = ((x + (-1.0 - x)) / (x * (-1.0 - x))) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9e+15)
		tmp = Float64(Float64(Float64(x + Float64(-1.0 - x)) / Float64(x * Float64(-1.0 - x))) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9e+15)
		tmp = ((x + (-1.0 - x)) / (x * (-1.0 - x))) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9e+15], N[(N[(N[(x + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e15

   1. Initial program 56.9%

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

      1. flip-- 56.2%

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

      2. div-inv 56.2%

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

      3. frac-times 56.2%

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

      4. metadata-eval 56.2%

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

      5. add-sqr-sqrt 57.1%

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

      6. frac-times 57.9%

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

      7. metadata-eval 57.9%

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

      8. add-sqr-sqrt 59.3%

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

      9. +-commutative 59.3%

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

      10. inv-pow 59.3%

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

      11. sqrt-pow2 59.3%

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

      12. metadata-eval 59.3%

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

      13. pow1/2 59.3%

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

      14. pow-flip 59.3%

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

      15. +-commutative 59.3%

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

      16. metadata-eval 59.3%

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

   4. Applied egg-rr 59.3%

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

      1. associate-*r/ 59.3%

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

      2. *-rgt-identity 59.3%

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

   6. Simplified 59.3%

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

      1. frac-2neg 59.3%

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

      2. metadata-eval 59.3%

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

      3. frac-sub 99.4%

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

      4. *-un-lft-identity 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. distribute-neg-in 99.4%

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

      8. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 9e15 < x

   1. Initial program 32.6%

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

      1. add-cube-cbrt 32.6%

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

      2. pow3 32.6%

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

      3. inv-pow 32.3%

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

      4. sqrt-pow2 25.7%

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

      5. metadata-eval 25.7%

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

      6. pow1/2 25.7%

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

      7. pow-flip 32.6%

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

      8. +-commutative 32.6%

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

      9. metadata-eval 32.6%

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

   4. Applied egg-rr 32.6%

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

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt[3]{0.5}\right)}}^{3} \]
   6. Step-by-step derivation

      1. *-commutative 98.7%

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

      2. unpow1/2 98.7%

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

      3. rem-exp-log 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5}\right)}^{3} \]

      4. exp-neg 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot {\color{blue}{\left(e^{-\log x}\right)}}^{0.5}\right)}^{3} \]

      5. exp-prod 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}}\right)}^{3} \]

      6. distribute-lft-neg-out 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\color{blue}{-\log x \cdot 0.5}}\right)}^{3} \]

      7. distribute-rgt-neg-in 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}}\right)}^{3} \]

      8. metadata-eval 93.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\log x \cdot \color{blue}{-0.5}}\right)}^{3} \]

      9. exp-to-pow 98.7%

         \[\leadsto {\left(\sqrt[3]{0.5} \cdot \color{blue}{{x}^{-0.5}}\right)}^{3} \]

   7. Simplified 98.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{0.5} \cdot {x}^{-0.5}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto {\color{blue}{\left({x}^{-0.5} \cdot \sqrt[3]{0.5}\right)}}^{3} \]

      2. unpow-prod-down 98.7%

         \[\leadsto \color{blue}{{\left({x}^{-0.5}\right)}^{3} \cdot {\left(\sqrt[3]{0.5}\right)}^{3}} \]

      3. pow-pow 98.6%

         \[\leadsto \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}} \cdot {\left(\sqrt[3]{0.5}\right)}^{3} \]

      4. metadata-eval 98.6%

         \[\leadsto {x}^{\color{blue}{-1.5}} \cdot {\left(\sqrt[3]{0.5}\right)}^{3} \]

      5. pow1/3 98.6%

         \[\leadsto {x}^{-1.5} \cdot {\color{blue}{\left({0.5}^{0.3333333333333333}\right)}}^{3} \]

      6. pow-pow 100.0%

         \[\leadsto {x}^{-1.5} \cdot \color{blue}{{0.5}^{\left(0.3333333333333333 \cdot 3\right)}} \]

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot {0.5}^{\color{blue}{1}} \]

      8. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot \color{blue}{0.5} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -1.5) 0.5))

C

double code(double x) {
	return pow(x, -1.5) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-1.5d0)) * 0.5d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.5) * 0.5;
}

Python

def code(x):
	return math.pow(x, -1.5) * 0.5

Julia

function code(x)
	return Float64((x ^ -1.5) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -1.5) * 0.5;
end

Wolfram

code[x_] := N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 34.0%

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

   1. add-cube-cbrt 34.0%

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

   2. pow3 34.0%

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

   3. inv-pow 33.6%

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

   4. sqrt-pow2 27.3%

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

   5. metadata-eval 27.3%

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

   6. pow1/2 27.3%

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

   7. pow-flip 34.0%

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

   8. +-commutative 34.0%

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

   9. metadata-eval 34.0%

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

4. Applied egg-rr 34.0%

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

5. Taylor expanded in x around inf 96.7%

   \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt[3]{0.5}\right)}}^{3} \]
6. Step-by-step derivation

   1. *-commutative 96.7%

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

   2. unpow1/2 96.7%

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

   3. rem-exp-log 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5}\right)}^{3} \]

   4. exp-neg 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot {\color{blue}{\left(e^{-\log x}\right)}}^{0.5}\right)}^{3} \]

   5. exp-prod 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}}\right)}^{3} \]

   6. distribute-lft-neg-out 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\color{blue}{-\log x \cdot 0.5}}\right)}^{3} \]

   7. distribute-rgt-neg-in 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}}\right)}^{3} \]

   8. metadata-eval 92.0%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot e^{\log x \cdot \color{blue}{-0.5}}\right)}^{3} \]

   9. exp-to-pow 96.7%

      \[\leadsto {\left(\sqrt[3]{0.5} \cdot \color{blue}{{x}^{-0.5}}\right)}^{3} \]

7. Simplified 96.7%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{0.5} \cdot {x}^{-0.5}\right)}}^{3} \]
8. Step-by-step derivation

   1. *-commutative 96.7%

      \[\leadsto {\color{blue}{\left({x}^{-0.5} \cdot \sqrt[3]{0.5}\right)}}^{3} \]

   2. unpow-prod-down 96.7%

      \[\leadsto \color{blue}{{\left({x}^{-0.5}\right)}^{3} \cdot {\left(\sqrt[3]{0.5}\right)}^{3}} \]

   3. pow-pow 96.7%

      \[\leadsto \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}} \cdot {\left(\sqrt[3]{0.5}\right)}^{3} \]

   4. metadata-eval 96.7%

      \[\leadsto {x}^{\color{blue}{-1.5}} \cdot {\left(\sqrt[3]{0.5}\right)}^{3} \]

   5. pow1/3 96.7%

      \[\leadsto {x}^{-1.5} \cdot {\color{blue}{\left({0.5}^{0.3333333333333333}\right)}}^{3} \]

   6. pow-pow 97.9%

      \[\leadsto {x}^{-1.5} \cdot \color{blue}{{0.5}^{\left(0.3333333333333333 \cdot 3\right)}} \]

   7. metadata-eval 97.9%

      \[\leadsto {x}^{-1.5} \cdot {0.5}^{\color{blue}{1}} \]

   8. metadata-eval 97.9%

      \[\leadsto {x}^{-1.5} \cdot \color{blue}{0.5} \]

9. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

10. Final simplification 97.9%

    \[\leadsto {x}^{-1.5} \cdot 0.5 \]
11. Add Preprocessing

Alternative 3: 5.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 34.0%

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

   1. expm1-log1p-u 34.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-undefine 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. inv-pow 5.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. sqrt-pow2 5.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 5.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 5.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. sub-neg 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} + \left(-1\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. log1p-undefine 5.3%

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

   3. rem-exp-log 5.3%

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

   4. +-commutative 5.3%

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

   5. metadata-eval 5.3%

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

   6. associate-+l+ 27.3%

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

   7. metadata-eval 27.3%

      \[\leadsto \left({x}^{-0.5} + \color{blue}{0}\right) - \frac{1}{\sqrt{x + 1}} \]

   8. +-rgt-identity 27.3%

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

6. Simplified 27.3%

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

7. Taylor expanded in x around 0 5.8%

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

   1. unpow1/2 5.8%

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

   2. rem-exp-log 5.8%

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

   3. exp-neg 5.8%

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

   4. exp-prod 5.8%

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

   5. distribute-lft-neg-out 5.8%

      \[\leadsto e^{\color{blue}{-\log x \cdot 0.5}} \]

   6. distribute-rgt-neg-in 5.8%

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

   7. metadata-eval 5.8%

      \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} \]

   8. exp-to-pow 5.8%

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

9. Simplified 5.8%

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

10. Final simplification 5.8%

    \[\leadsto {x}^{-0.5} \]
11. Add Preprocessing

Developer target: 98.4% 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 2024060 
(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)))))