2isqrt (example 3.6)

Percentage Accurate: 38.0% → 99.9%

Time: 14.6s

Alternatives: 5

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: 38.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x} \cdot \frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* (pow x -1.5) (+ 0.5 (/ 0.5 x)))
   (/
    1.0
    (/
     (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))
     (* (/ 1.0 (+ 1.0 x)) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = pow(x, -1.5) * (0.5 + (0.5 / x));
	} else {
		tmp = 1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) / ((1.0 / (1.0 + x)) * (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) * (0.5d0 + (0.5d0 / x))
    else
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) / ((1.0d0 / (1.0d0 + x)) * (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = Math.pow(x, -1.5) * (0.5 + (0.5 / x));
	} else {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) / ((1.0 / (1.0 + x)) * (1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = math.pow(x, -1.5) * (0.5 + (0.5 / x))
	else:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) / ((1.0 / (1.0 + x)) * (1.0 / x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.9%

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

   3. Taylor expanded in x around inf 82.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. div-sub 82.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

      2. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

      3. associate-/l* 82.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

      4. *-commutative 82.9%

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \frac{\color{blue}{\sqrt{x} \cdot -0.5}}{{x}^{2}} \]

      5. associate-/l* 82.9%

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \color{blue}{\sqrt{x} \cdot \frac{-0.5}{{x}^{2}}} \]

      6. distribute-rgt-out-- 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

      2. distribute-rgt-out 99.7%

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

      3. associate-/l* 99.7%

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

      4. pow-flip 99.7%

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

      5. sqrt-pow1 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{\color{blue}{-1.5}} + \sqrt{\frac{1}{x}}}{x}\right) \]

      8. inv-pow 99.7%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \sqrt{\color{blue}{{x}^{-1}}}}{x}\right) \]

      9. sqrt-pow1 99.8%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x}\right) \]

      10. metadata-eval 99.8%

          \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{\color{blue}{-0.5}}}{x}\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 99.8%

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

   12. Simplified 99.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}} \]
   13. Step-by-step derivation

       1. associate-*r/ 99.8%

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

       2. associate-*l/ 99.7%

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

       3. distribute-rgt-in 99.7%

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

       4. *-commutative 99.7%

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

       5. div-inv 99.7%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot {x}^{-0.5} \]

       6. associate-*l* 99.7%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot {x}^{-0.5}\right)} \]

       7. inv-pow 99.7%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot \left(\color{blue}{{x}^{-1}} \cdot {x}^{-0.5}\right) \]

       8. metadata-eval 99.7%

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

       9. pow-prod-up 99.3%

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

       10. unpow3 99.3%

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

       11. pow-pow 100.0%

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

       12. metadata-eval 100.0%

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

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot {x}^{-1.5}} \]
   15. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. distribute-lft-out 100.0%

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

   16. Simplified 100.0%

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

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

   1. Initial program 56.1%

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

      1. flip-- 55.2%

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

      2. clear-num 55.2%

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

      3. inv-pow 55.2%

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

      4. sqrt-pow2 55.2%

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

      5. metadata-eval 55.2%

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

      6. inv-pow 55.2%

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

      7. sqrt-pow2 55.2%

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

      8. +-commutative 55.2%

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

      9. metadata-eval 55.2%

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

      10. frac-times 56.3%

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

      11. metadata-eval 56.3%

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

      12. add-sqr-sqrt 55.9%

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

      13. frac-times 56.9%

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

      14. metadata-eval 56.9%

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

      15. add-sqr-sqrt 57.9%

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

      16. +-commutative 57.9%

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

   4. Applied egg-rr 57.9%

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

      1. frac-sub 93.0%

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

      2. *-un-lft-identity 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. *-un-lft-identity 93.0%

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

      2. *-commutative 93.0%

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

      3. times-frac 93.2%

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

      4. +-commutative 93.2%

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

      5. *-rgt-identity 93.2%

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

      6. associate--l+ 99.2%

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

      7. +-inverses 99.2%

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

      8. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x} \cdot \frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + x\right) \cdot \left(x \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-24)
   (* (pow x -1.5) (+ 0.5 (/ 0.5 x)))
   (/ 1.0 (* (+ 1.0 x) (* x (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-24) {
		tmp = pow(x, -1.5) * (0.5 + (0.5 / x));
	} else {
		tmp = 1.0 / ((1.0 + x) * (x * (pow(x, -0.5) + pow((1.0 + x), -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-24) then
        tmp = (x ** (-1.5d0)) * (0.5d0 + (0.5d0 / x))
    else
        tmp = 1.0d0 / ((1.0d0 + x) * (x * ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-24) {
		tmp = Math.pow(x, -1.5) * (0.5 + (0.5 / x));
	} else {
		tmp = 1.0 / ((1.0 + x) * (x * (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-24:
		tmp = math.pow(x, -1.5) * (0.5 + (0.5 / x))
	else:
		tmp = 1.0 / ((1.0 + x) * (x * (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-24)
		tmp = Float64((x ^ -1.5) * Float64(0.5 + Float64(0.5 / x)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + x) * Float64(x * Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-24)
		tmp = (x ^ -1.5) * (0.5 + (0.5 / x));
	else
		tmp = 1.0 / ((1.0 + x) * (x * ((x ^ -0.5) + ((1.0 + x) ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-24], N[(N[Power[x, -1.5], $MachinePrecision] * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(x * N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 1.99999999999999985e-24

   1. Initial program 42.8%

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

   3. Taylor expanded in x around inf 83.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. div-sub 83.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

      2. *-commutative 83.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

      4. *-commutative 83.0%

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \frac{\color{blue}{\sqrt{x} \cdot -0.5}}{{x}^{2}} \]

      5. associate-/l* 83.0%

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \color{blue}{\sqrt{x} \cdot \frac{-0.5}{{x}^{2}}} \]

      6. distribute-rgt-out-- 83.0%

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

   5. Simplified 83.0%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

      2. distribute-rgt-out 99.7%

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

      3. associate-/l* 99.7%

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

      4. pow-flip 99.7%

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

      5. sqrt-pow1 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{\color{blue}{-1.5}} + \sqrt{\frac{1}{x}}}{x}\right) \]

      8. inv-pow 99.7%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \sqrt{\color{blue}{{x}^{-1}}}}{x}\right) \]

      9. sqrt-pow1 99.7%

         \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x}\right) \]

      10. metadata-eval 99.7%

          \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{\color{blue}{-0.5}}}{x}\right) \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 99.7%

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

   12. Simplified 99.7%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}} \]
   13. Step-by-step derivation

       1. associate-*r/ 99.8%

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

       2. associate-*l/ 99.6%

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

       3. distribute-rgt-in 99.6%

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

       4. *-commutative 99.6%

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

       5. div-inv 99.6%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot {x}^{-0.5} \]

       6. associate-*l* 99.6%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot {x}^{-0.5}\right)} \]

       7. inv-pow 99.6%

          \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot \left(\color{blue}{{x}^{-1}} \cdot {x}^{-0.5}\right) \]

       8. metadata-eval 99.6%

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

       9. pow-prod-up 99.3%

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

       10. unpow3 99.3%

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

       11. pow-pow 100.0%

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

       12. metadata-eval 100.0%

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

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot {x}^{-1.5}} \]
   15. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. distribute-lft-out 100.0%

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

   16. Simplified 100.0%

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

   if 1.99999999999999985e-24 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 59.1%

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

      1. flip-- 58.1%

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

      2. clear-num 58.1%

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

      3. inv-pow 58.1%

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

      4. sqrt-pow2 58.1%

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

      5. metadata-eval 58.1%

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

      6. inv-pow 58.1%

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

      7. sqrt-pow2 58.1%

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

      8. +-commutative 58.1%

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

      9. metadata-eval 58.1%

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

      10. frac-times 59.2%

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

      11. metadata-eval 59.2%

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

      12. add-sqr-sqrt 58.8%

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

      13. frac-times 60.0%

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

      14. metadata-eval 60.0%

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

      15. add-sqr-sqrt 61.0%

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

      16. +-commutative 61.0%

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

   4. Applied egg-rr 61.0%

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

      1. frac-sub 98.7%

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

      2. *-un-lft-identity 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. associate-/r/ 99.1%

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

      2. associate-*r* 99.0%

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

      3. flip-+ 61.1%

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

      4. *-rgt-identity 61.1%

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

      5. associate--l+ 61.2%

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

      6. +-inverses 61.2%

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

      7. metadata-eval 61.2%

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

      8. associate-/l/ 61.2%

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

      9. *-un-lft-identity 61.2%

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

      10. flip-+ 99.3%

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

      11. +-commutative 99.3%

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

      12. +-commutative 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + x\right) \cdot \left(x \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.7%

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

3. Taylor expanded in x around inf 82.0%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. div-sub 82.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

   2. *-commutative 82.0%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

   3. associate-/l* 82.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

   4. *-commutative 82.0%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \frac{\color{blue}{\sqrt{x} \cdot -0.5}}{{x}^{2}} \]

   5. associate-/l* 81.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \color{blue}{\sqrt{x} \cdot \frac{-0.5}{{x}^{2}}} \]

   6. distribute-rgt-out-- 81.9%

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

5. Simplified 81.9%

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

6. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 0.0%

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

8. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 97.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

   2. distribute-rgt-out 97.6%

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

   3. associate-/l* 97.6%

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

   4. pow-flip 97.6%

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

   5. sqrt-pow1 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{\color{blue}{-1.5}} + \sqrt{\frac{1}{x}}}{x}\right) \]

   8. inv-pow 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \sqrt{\color{blue}{{x}^{-1}}}}{x}\right) \]

   9. sqrt-pow1 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x}\right) \]

   10. metadata-eval 97.6%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{\color{blue}{-0.5}}}{x}\right) \]

10. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 97.6%

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

12. Simplified 97.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}} \]
13. Step-by-step derivation

    1. associate-*r/ 97.7%

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

    2. associate-*l/ 97.5%

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

    3. distribute-rgt-in 97.5%

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

    4. *-commutative 97.5%

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

    5. div-inv 97.5%

       \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot {x}^{-0.5} \]

    6. associate-*l* 97.5%

       \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot {x}^{-0.5}\right)} \]

    7. inv-pow 97.5%

       \[\leadsto {x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot \left(\color{blue}{{x}^{-1}} \cdot {x}^{-0.5}\right) \]

    8. metadata-eval 97.5%

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

    9. pow-prod-up 97.2%

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

    10. unpow3 97.2%

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

    11. pow-pow 97.9%

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

    12. metadata-eval 97.9%

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

14. Applied egg-rr 97.9%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot \frac{0.5}{x} + 0.5 \cdot {x}^{-1.5}} \]
15. Step-by-step derivation

    1. *-commutative 97.9%

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

    2. distribute-lft-out 97.9%

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

16. Simplified 97.9%

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

17. Final simplification 97.9%

    \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{0.5}{x}\right) \]
18. Add Preprocessing

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

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(0.5 * Float64(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[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.7%

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

3. Taylor expanded in x around inf 82.0%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. div-sub 82.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

   2. *-commutative 82.0%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot -0.5}}{{x}^{2}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

   3. associate-/l* 82.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}}} - \frac{-0.5 \cdot \sqrt{x}}{{x}^{2}} \]

   4. *-commutative 82.0%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \frac{\color{blue}{\sqrt{x} \cdot -0.5}}{{x}^{2}} \]

   5. associate-/l* 81.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{{x}^{2}} - \color{blue}{\sqrt{x} \cdot \frac{-0.5}{{x}^{2}}} \]

   6. distribute-rgt-out-- 81.9%

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

5. Simplified 81.9%

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

6. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 0.0%

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

8. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 97.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5 + \sqrt{\frac{1}{x}} \cdot 0.5}{x}} \]

   2. distribute-rgt-out 97.6%

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

   3. associate-/l* 97.6%

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

   4. pow-flip 97.6%

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

   5. sqrt-pow1 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{\color{blue}{-1.5}} + \sqrt{\frac{1}{x}}}{x}\right) \]

   8. inv-pow 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \sqrt{\color{blue}{{x}^{-1}}}}{x}\right) \]

   9. sqrt-pow1 97.6%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x}\right) \]

   10. metadata-eval 97.6%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{\color{blue}{-0.5}}}{x}\right) \]

10. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \frac{{x}^{-1.5} + {x}^{-0.5}}{x}\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 97.6%

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

12. Simplified 97.6%

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

13. Taylor expanded in x around inf 97.7%

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

14. Final simplification 97.7%

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

Alternative 5: 44.0% 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 43.7%

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

   1. flip-- 43.6%

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

   2. clear-num 43.6%

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

   3. inv-pow 43.6%

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

   4. sqrt-pow2 43.6%

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

   5. metadata-eval 43.6%

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

   6. inv-pow 43.6%

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

   7. sqrt-pow2 43.6%

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

   8. +-commutative 43.6%

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

   9. metadata-eval 43.6%

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

   10. frac-times 23.8%

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

   11. metadata-eval 23.8%

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

   12. add-sqr-sqrt 21.6%

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

   13. frac-times 26.0%

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

   14. metadata-eval 26.0%

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

   15. add-sqr-sqrt 43.8%

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

   16. +-commutative 43.8%

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

4. Applied egg-rr 43.8%

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

5. Taylor expanded in x around inf 81.7%

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

6. Taylor expanded in x around 0 45.1%

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

   1. metadata-eval 45.1%

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

   2. cube-div 45.2%

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

8. Simplified 45.2%

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

   1. sqrt-pow1 48.8%

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

   2. inv-pow 48.8%

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

   3. pow-pow 48.8%

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

   4. metadata-eval 48.8%

      \[\leadsto {x}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

   5. metadata-eval 48.8%

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

   6. *-un-lft-identity 48.8%

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

   7. *-commutative 48.8%

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

10. Applied egg-rr 48.8%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot 1} \]
11. Step-by-step derivation

    1. *-rgt-identity 48.8%

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

12. Simplified 48.8%

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

13. Final simplification 48.8%

    \[\leadsto {x}^{-1.5} \]
14. 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 2024075 
(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)))))