2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.0%

Time: 14.3s

Alternatives: 6

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: 38.8% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ 0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (* 0.3125 (pow x -3.5))
  (fma -0.5 (/ (pow x -0.5) x) (* (/ (pow x -0.5) (pow x 2.0)) 0.375))))

C

double code(double x) {
	return (0.3125 * pow(x, -3.5)) - fma(-0.5, (pow(x, -0.5) / x), ((pow(x, -0.5) / pow(x, 2.0)) * 0.375));
}

Julia

function code(x)
	return Float64(Float64(0.3125 * (x ^ -3.5)) - fma(-0.5, Float64((x ^ -0.5) / x), Float64(Float64((x ^ -0.5) / (x ^ 2.0)) * 0.375)))
end

Wolfram

code[x_] := N[(N[(0.3125 * N[Power[x, -3.5], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Power[x, -0.5], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.0%

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

   1. add-exp-log 7.4%

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

   2. log-rec 7.4%

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

   3. pow1/2 7.4%

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

   4. log-pow 7.4%

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

   5. +-commutative 7.4%

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

   6. log1p-define 7.4%

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

4. Applied egg-rr 7.4%

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

5. Taylor expanded in x around inf 7.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} + 0.3125 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{3}}\right) - \left(e^{--0.5 \cdot \log \left(\frac{1}{x}\right)} + \left(-0.5 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{x} + 0.375 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 7.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\frac{1}{x}} + 0.3125 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{3}}\right) - e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}\right) - \left(-0.5 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{x} + 0.375 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{2}}\right)} \]

7. Simplified 99.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3125, \frac{{x}^{-0.5}}{{x}^{3}}, {x}^{-0.5}\right) - {x}^{-0.5}\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)} \]
8. Step-by-step derivation

   1. sub-neg 99.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3125, \frac{{x}^{-0.5}}{{x}^{3}}, {x}^{-0.5}\right) + \left(-{x}^{-0.5}\right)\right)} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

   2. pow-div 99.0%

      \[\leadsto \left(\mathsf{fma}\left(0.3125, \color{blue}{{x}^{\left(-0.5 - 3\right)}}, {x}^{-0.5}\right) + \left(-{x}^{-0.5}\right)\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

   3. metadata-eval 99.0%

      \[\leadsto \left(\mathsf{fma}\left(0.3125, {x}^{\color{blue}{-3.5}}, {x}^{-0.5}\right) + \left(-{x}^{-0.5}\right)\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

9. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3125, {x}^{-3.5}, {x}^{-0.5}\right) + \left(-{x}^{-0.5}\right)\right)} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]
10. Step-by-step derivation

    1. fma-undefine 99.0%

       \[\leadsto \left(\color{blue}{\left(0.3125 \cdot {x}^{-3.5} + {x}^{-0.5}\right)} + \left(-{x}^{-0.5}\right)\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

    2. associate-+r+ 99.4%

       \[\leadsto \color{blue}{\left(0.3125 \cdot {x}^{-3.5} + \left({x}^{-0.5} + \left(-{x}^{-0.5}\right)\right)\right)} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

    3. sub-neg 99.4%

       \[\leadsto \left(0.3125 \cdot {x}^{-3.5} + \color{blue}{\left({x}^{-0.5} - {x}^{-0.5}\right)}\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

    4. +-inverses 99.4%

       \[\leadsto \left(0.3125 \cdot {x}^{-3.5} + \color{blue}{0}\right) - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

    5. +-commutative 99.4%

       \[\leadsto \color{blue}{\left(0 + 0.3125 \cdot {x}^{-3.5}\right)} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

    6. +-lft-identity 99.4%

       \[\leadsto \color{blue}{0.3125 \cdot {x}^{-3.5}} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

11. Simplified 99.4%

    \[\leadsto \color{blue}{0.3125 \cdot {x}^{-3.5}} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

12. Final simplification 99.4%

    \[\leadsto 0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, \frac{{x}^{-0.5}}{x}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]
13. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{0.375}{x} + \frac{0.2734375}{{x}^{3}}\right)}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (- (+ 0.5 (/ 0.3125 (pow x 2.0))) (+ (/ 0.375 x) (/ 0.2734375 (pow x 3.0))))
   x)
  (sqrt x)))

C

double code(double x) {
	return (((0.5 + (0.3125 / pow(x, 2.0))) - ((0.375 / x) + (0.2734375 / pow(x, 3.0)))) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 + (0.3125d0 / (x ** 2.0d0))) - ((0.375d0 / x) + (0.2734375d0 / (x ** 3.0d0)))) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return (((0.5 + (0.3125 / Math.pow(x, 2.0))) - ((0.375 / x) + (0.2734375 / Math.pow(x, 3.0)))) / x) / Math.sqrt(x);
}

Python

def code(x):
	return (((0.5 + (0.3125 / math.pow(x, 2.0))) - ((0.375 / x) + (0.2734375 / math.pow(x, 3.0)))) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.3125 / (x ^ 2.0))) - Float64(Float64(0.375 / x) + Float64(0.2734375 / (x ^ 3.0)))) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 + (0.3125 / (x ^ 2.0))) - ((0.375 / x) + (0.2734375 / (x ^ 3.0)))) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.3125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 / x), $MachinePrecision] + N[(0.2734375 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.0%

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

   1. expm1-log1p-u 46.0%

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

   2. expm1-undefine 5.6%

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

   3. inv-pow 5.6%

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

   4. sqrt-pow2 5.6%

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

   5. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

6. Simplified 33.1%

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

   1. metadata-eval 33.1%

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

   2. pow-flip 45.7%

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

   3. pow1/2 46.0%

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

   4. clear-num 46.0%

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

   5. +-commutative 46.0%

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

   6. frac-sub 46.1%

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

   7. *-un-lft-identity 46.1%

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

8. Applied egg-rr 46.1%

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

   1. div-sub 46.0%

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

   2. /-rgt-identity 46.0%

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

   3. *-rgt-identity 46.0%

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

   4. /-rgt-identity 46.0%

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

   5. /-rgt-identity 46.0%

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

   6. div-sub 46.1%

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

   7. rem-square-sqrt 46.1%

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

   8. times-frac 46.1%

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

   9. associate-*l/ 46.1%

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

10. Simplified 46.1%

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

11. Taylor expanded in x around inf 99.3%

    \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(0.375 \cdot \frac{1}{x} + 0.2734375 \cdot \frac{1}{{x}^{3}}\right)}{x}}}{\sqrt{x}} \]
12. Step-by-step derivation

    1. associate-*r/ 99.3%

       \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\color{blue}{\frac{0.375 \cdot 1}{x}} + 0.2734375 \cdot \frac{1}{{x}^{3}}\right)}{x}}{\sqrt{x}} \]

    2. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{\color{blue}{0.375}}{x} + 0.2734375 \cdot \frac{1}{{x}^{3}}\right)}{x}}{\sqrt{x}} \]

    3. associate-*r/ 99.3%

       \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{0.375}{x} + \color{blue}{\frac{0.2734375 \cdot 1}{{x}^{3}}}\right)}{x}}{\sqrt{x}} \]

    4. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{0.375}{x} + \frac{\color{blue}{0.2734375}}{{x}^{3}}\right)}{x}}{\sqrt{x}} \]

13. Simplified 99.3%

    \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{0.375}{x} + \frac{0.2734375}{{x}^{3}}\right)}{x}}}{\sqrt{x}} \]

14. Final simplification 99.3%

    \[\leadsto \frac{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - \left(\frac{0.375}{x} + \frac{0.2734375}{{x}^{3}}\right)}{x}}{\sqrt{x}} \]
15. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \left(\frac{0.3125}{{x}^{2}} - \frac{0.375}{x}\right)}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ 0.5 (- (/ 0.3125 (pow x 2.0)) (/ 0.375 x))) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 + ((0.3125 / pow(x, 2.0)) - (0.375 / x))) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + ((0.3125d0 / (x ** 2.0d0)) - (0.375d0 / x))) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 + ((0.3125 / Math.pow(x, 2.0)) - (0.375 / x))) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 + ((0.3125 / math.pow(x, 2.0)) - (0.375 / x))) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(0.3125 / (x ^ 2.0)) - Float64(0.375 / x))) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + ((0.3125 / (x ^ 2.0)) - (0.375 / x))) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 + N[(N[(0.3125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.0%

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

   1. expm1-log1p-u 46.0%

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

   2. expm1-undefine 5.6%

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

   3. inv-pow 5.6%

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

   4. sqrt-pow2 5.6%

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

   5. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

6. Simplified 33.1%

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

   1. metadata-eval 33.1%

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

   2. pow-flip 45.7%

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

   3. pow1/2 46.0%

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

   4. clear-num 46.0%

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

   5. +-commutative 46.0%

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

   6. frac-sub 46.1%

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

   7. *-un-lft-identity 46.1%

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

8. Applied egg-rr 46.1%

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

   1. div-sub 46.0%

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

   2. /-rgt-identity 46.0%

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

   3. *-rgt-identity 46.0%

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

   4. /-rgt-identity 46.0%

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

   5. /-rgt-identity 46.0%

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

   6. div-sub 46.1%

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

   7. rem-square-sqrt 46.1%

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

   8. times-frac 46.1%

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

   9. associate-*l/ 46.1%

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

10. Simplified 46.1%

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

11. Taylor expanded in x around inf 99.2%

    \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + \frac{0.3125}{{x}^{2}}\right) - 0.375 \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
12. Step-by-step derivation

    1. associate--l+ 99.2%

       \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(\frac{0.3125}{{x}^{2}} - 0.375 \cdot \frac{1}{x}\right)}}{x}}{\sqrt{x}} \]

    2. associate-*r/ 99.2%

       \[\leadsto \frac{\frac{0.5 + \left(\frac{0.3125}{{x}^{2}} - \color{blue}{\frac{0.375 \cdot 1}{x}}\right)}{x}}{\sqrt{x}} \]

    3. metadata-eval 99.2%

       \[\leadsto \frac{\frac{0.5 + \left(\frac{0.3125}{{x}^{2}} - \frac{\color{blue}{0.375}}{x}\right)}{x}}{\sqrt{x}} \]

13. Simplified 99.2%

    \[\leadsto \frac{\color{blue}{\frac{0.5 + \left(\frac{0.3125}{{x}^{2}} - \frac{0.375}{x}\right)}{x}}}{\sqrt{x}} \]

14. Final simplification 99.2%

    \[\leadsto \frac{\frac{0.5 + \left(\frac{0.3125}{{x}^{2}} - \frac{0.375}{x}\right)}{x}}{\sqrt{x}} \]
15. Add Preprocessing

Alternative 4: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.0%

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

   1. expm1-log1p-u 46.0%

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

   2. expm1-undefine 5.6%

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

   3. inv-pow 5.6%

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

   4. sqrt-pow2 5.6%

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

   5. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

6. Simplified 33.1%

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

   1. metadata-eval 33.1%

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

   2. pow-flip 45.7%

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

   3. pow1/2 46.0%

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

   4. clear-num 46.0%

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

   5. +-commutative 46.0%

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

   6. frac-sub 46.1%

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

   7. *-un-lft-identity 46.1%

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

8. Applied egg-rr 46.1%

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

   1. div-sub 46.0%

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

   2. /-rgt-identity 46.0%

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

   3. *-rgt-identity 46.0%

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

   4. /-rgt-identity 46.0%

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

   5. /-rgt-identity 46.0%

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

   6. div-sub 46.1%

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

   7. rem-square-sqrt 46.1%

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

   8. times-frac 46.1%

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

   9. associate-*l/ 46.1%

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

10. Simplified 46.1%

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

11. Taylor expanded in x around inf 98.8%

    \[\leadsto \frac{\color{blue}{\frac{0.5 - 0.375 \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
12. Step-by-step derivation

    1. associate-*r/ 98.8%

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

    2. metadata-eval 98.8%

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

    \[\leadsto \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \]
15. Add Preprocessing

Alternative 5: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.0%

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

   1. frac-sub 46.1%

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

   2. *-un-lft-identity 46.1%

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

   3. *-rgt-identity 46.1%

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

   4. +-commutative 46.1%

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

   5. sqrt-unprod 46.1%

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

   6. +-commutative 46.1%

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

4. Applied egg-rr 46.1%

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

5. Taylor expanded in x around inf 45.3%

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

6. Taylor expanded in x around inf 70.0%

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

   1. *-commutative 70.0%

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

8. Simplified 70.0%

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

   1. *-un-lft-identity 70.0%

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

   2. pow-flip 70.6%

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

   3. sqrt-pow1 97.8%

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

   4. metadata-eval 97.8%

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

   5. metadata-eval 97.8%

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

10. Applied egg-rr 97.8%

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

    1. *-lft-identity 97.8%

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

12. Simplified 97.8%

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

13. Final simplification 97.8%

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

Alternative 6: 5.6% 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 46.0%

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

   1. expm1-log1p-u 46.0%

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

   2. expm1-undefine 5.6%

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

   3. inv-pow 5.6%

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

   4. sqrt-pow2 5.6%

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

   5. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

6. Simplified 33.1%

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

7. Taylor expanded in x around 0 5.5%

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

   1. unpow1/2 5.5%

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

   2. exp-to-pow 5.5%

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

   3. *-commutative 5.5%

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

   4. *-commutative 5.5%

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

   5. log-rec 5.5%

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

   6. distribute-lft-neg-out 5.5%

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

   7. distribute-rgt-neg-in 5.5%

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

   8. metadata-eval 5.5%

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

   9. exp-to-pow 5.5%

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

9. Simplified 5.5%

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

10. Final simplification 5.5%

    \[\leadsto {x}^{-0.5} \]
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 2024112 
(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)))))