2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.0%

Time: 15.1s

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: 38.4% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + {x}^{-2.5} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (+ (* 0.5 (sqrt (/ 1.0 x))) (* (pow x -2.5) 0.3125))
   (* (sqrt (pow x -3.0)) -0.375))
  x))

C

double code(double x) {
	return (((0.5 * sqrt((1.0 / x))) + (pow(x, -2.5) * 0.3125)) + (sqrt(pow(x, -3.0)) * -0.375)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 * sqrt((1.0d0 / x))) + ((x ** (-2.5d0)) * 0.3125d0)) + (sqrt((x ** (-3.0d0))) * (-0.375d0))) / x
end function

Java

public static double code(double x) {
	return (((0.5 * Math.sqrt((1.0 / x))) + (Math.pow(x, -2.5) * 0.3125)) + (Math.sqrt(Math.pow(x, -3.0)) * -0.375)) / x;
}

Python

def code(x):
	return (((0.5 * math.sqrt((1.0 / x))) + (math.pow(x, -2.5) * 0.3125)) + (math.sqrt(math.pow(x, -3.0)) * -0.375)) / x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64((x ^ -2.5) * 0.3125)) + Float64(sqrt((x ^ -3.0)) * -0.375)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 * sqrt((1.0 / x))) + ((x ^ -2.5) * 0.3125)) + (sqrt((x ^ -3.0)) * -0.375)) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -2.5], $MachinePrecision] * 0.3125), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

   1. expm1-log1p-u 39.0%

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

   2. expm1-undefine 4.7%

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

   3. inv-pow 4.7%

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

   4. sqrt-pow2 4.7%

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

   5. metadata-eval 4.7%

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

4. Applied egg-rr 4.7%

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

   1. sub-neg 4.7%

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

   2. log1p-undefine 4.7%

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

   3. rem-exp-log 4.7%

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

   4. +-commutative 4.7%

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

   5. metadata-eval 4.7%

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

   6. associate-+l+ 31.1%

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

   7. metadata-eval 31.1%

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

   8. +-rgt-identity 31.1%

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

6. Simplified 31.1%

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

   1. metadata-eval 31.1%

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

   2. pow-pow 22.8%

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

   3. sqr-pow 21.7%

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

   4. add-sqr-sqrt 32.1%

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

   5. difference-of-squares 32.1%

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

   6. metadata-eval 32.1%

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

   7. pow-pow 32.1%

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

   8. metadata-eval 32.1%

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

   9. inv-pow 32.1%

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

   10. +-commutative 32.1%

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

   11. sqrt-pow2 32.1%

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

   12. metadata-eval 32.1%

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

   13. sqrt-pow1 32.1%

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

   14. metadata-eval 32.1%

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

8. Applied egg-rr 39.1%

   \[\leadsto \color{blue}{\left({x}^{-0.25} + {\left(1 + x\right)}^{-0.25}\right) \cdot \left({x}^{-0.25} - {\left(1 + x\right)}^{-0.25}\right)} \]

9. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{\frac{-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.078125 \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(0.234375 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{x}} \]
10. Step-by-step derivation

11. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

    2. pow1/2 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

    3. pow-flip 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

    4. pow-pow 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

    5. metadata-eval 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

    6. metadata-eval 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

13. Applied egg-rr 99.4%

    \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]
14. Step-by-step derivation

    1. *-lft-identity 99.4%

       \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{{x}^{-2.5}} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

15. Simplified 99.4%

    \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{{x}^{-2.5}} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]

16. Final simplification 99.4%

    \[\leadsto \frac{\left(0.5 \cdot \sqrt{\frac{1}{x}} + {x}^{-2.5} \cdot 0.3125\right) + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]
17. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{{x}^{-3}} \cdot -0.375}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* 0.5 (sqrt (/ 1.0 x))) (* (sqrt (pow x -3.0)) -0.375)) x))

C

double code(double x) {
	return ((0.5 * sqrt((1.0 / x))) + (sqrt(pow(x, -3.0)) * -0.375)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 * sqrt((1.0d0 / x))) + (sqrt((x ** (-3.0d0))) * (-0.375d0))) / x
end function

Java

public static double code(double x) {
	return ((0.5 * Math.sqrt((1.0 / x))) + (Math.sqrt(Math.pow(x, -3.0)) * -0.375)) / x;
}

Python

def code(x):
	return ((0.5 * math.sqrt((1.0 / x))) + (math.sqrt(math.pow(x, -3.0)) * -0.375)) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(sqrt((x ^ -3.0)) * -0.375)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 * sqrt((1.0 / x))) + (sqrt((x ^ -3.0)) * -0.375)) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

   1. expm1-log1p-u 39.0%

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

   2. expm1-undefine 4.7%

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

   3. inv-pow 4.7%

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

   4. sqrt-pow2 4.7%

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

   5. metadata-eval 4.7%

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

4. Applied egg-rr 4.7%

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

   1. sub-neg 4.7%

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

   2. log1p-undefine 4.7%

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

   3. rem-exp-log 4.7%

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

   4. +-commutative 4.7%

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

   5. metadata-eval 4.7%

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

   6. associate-+l+ 31.1%

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

   7. metadata-eval 31.1%

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

   8. +-rgt-identity 31.1%

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

6. Simplified 31.1%

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

   1. metadata-eval 31.1%

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

   2. pow-pow 22.8%

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

   3. sqr-pow 21.7%

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

   4. add-sqr-sqrt 32.1%

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

   5. difference-of-squares 32.1%

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

   6. metadata-eval 32.1%

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

   7. pow-pow 32.1%

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

   8. metadata-eval 32.1%

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

   9. inv-pow 32.1%

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

   10. +-commutative 32.1%

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

   11. sqrt-pow2 32.1%

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

   12. metadata-eval 32.1%

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

   13. sqrt-pow1 32.1%

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

   14. metadata-eval 32.1%

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

8. Applied egg-rr 39.1%

   \[\leadsto \color{blue}{\left({x}^{-0.25} + {\left(1 + x\right)}^{-0.25}\right) \cdot \left({x}^{-0.25} - {\left(1 + x\right)}^{-0.25}\right)} \]

9. Taylor expanded in x around inf 99.2%

   \[\leadsto \color{blue}{\frac{-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
10. Step-by-step derivation

    1. associate-+r+ 99.2%

       \[\leadsto \frac{\color{blue}{\left(-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + -0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + 0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]

    2. +-commutative 99.2%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + \left(-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + -0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}}{x} \]

    3. distribute-rgt-out 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \left(-0.3125 + -0.0625\right)}}{x} \]

    4. exp-to-pow 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    5. *-commutative 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    6. rec-exp 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    7. mul-1-neg 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    8. associate-*r* 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    9. metadata-eval 99.2%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    10. *-commutative 99.2%

        \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    11. exp-to-pow 99.2%

        \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\color{blue}{{x}^{-3}}} \cdot \left(-0.3125 + -0.0625\right)}{x} \]

    12. metadata-eval 99.2%

        \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{{x}^{-3}} \cdot \color{blue}{-0.375}}{x} \]

11. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{{x}^{-3}} \cdot -0.375}{x}} \]

12. Final simplification 99.2%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{{x}^{-3}} \cdot -0.375}{x} \]
13. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \left({x}^{-2.5} - {x}^{-1.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -0.5 (- (pow x -2.5) (pow x -1.5))))

C

double code(double x) {
	return -0.5 * (pow(x, -2.5) - pow(x, -1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) * ((x ** (-2.5d0)) - (x ** (-1.5d0)))
end function

Java

public static double code(double x) {
	return -0.5 * (Math.pow(x, -2.5) - Math.pow(x, -1.5));
}

Python

def code(x):
	return -0.5 * (math.pow(x, -2.5) - math.pow(x, -1.5))

Julia

function code(x)
	return Float64(-0.5 * Float64((x ^ -2.5) - (x ^ -1.5)))
end

MATLAB

function tmp = code(x)
	tmp = -0.5 * ((x ^ -2.5) - (x ^ -1.5));
end

Wolfram

code[x_] := N[(-0.5 * N[(N[Power[x, -2.5], $MachinePrecision] - N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

   1. expm1-log1p-u 39.0%

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

   2. expm1-undefine 4.7%

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

   3. inv-pow 4.7%

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

   4. sqrt-pow2 4.7%

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

   5. metadata-eval 4.7%

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

4. Applied egg-rr 4.7%

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

   1. sub-neg 4.7%

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

   2. log1p-undefine 4.7%

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

   3. rem-exp-log 4.7%

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

   4. +-commutative 4.7%

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

   5. metadata-eval 4.7%

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

   6. associate-+l+ 31.1%

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

   7. metadata-eval 31.1%

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

   8. +-rgt-identity 31.1%

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

6. Simplified 31.1%

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

7. Taylor expanded in x around inf 85.0%

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

   1. distribute-lft-out-- 85.0%

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

   2. associate-/l* 85.0%

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

   3. unpow1/2 85.0%

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

   4. exp-to-pow 85.0%

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

   5. log-rec 85.0%

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

   6. distribute-lft-neg-out 85.0%

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

   7. distribute-rgt-neg-in 85.0%

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

   8. metadata-eval 85.0%

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

   9. exp-to-pow 85.0%

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

9. Simplified 85.0%

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

    1. div-sub 85.0%

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

    2. pow1/2 85.0%

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

    3. pow-div 98.7%

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

    4. metadata-eval 98.7%

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

    5. metadata-eval 98.7%

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

    6. sqrt-pow1 71.3%

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

    7. pow-div 71.3%

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

    8. metadata-eval 71.3%

       \[\leadsto -0.5 \cdot \left({x}^{\color{blue}{-2.5}} - \sqrt{{x}^{-3}}\right) \]

    9. metadata-eval 71.3%

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

    10. metadata-eval 71.3%

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

    11. sqrt-pow1 71.3%

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

    12. pow-flip 71.3%

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

    13. sub-neg 71.3%

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

    14. pow-flip 71.3%

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

    15. sqrt-pow1 71.3%

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

    16. metadata-eval 71.3%

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

    17. metadata-eval 71.3%

        \[\leadsto -0.5 \cdot \left({x}^{\color{blue}{-2.5}} + \left(-\sqrt{{x}^{-3}}\right)\right) \]

    18. sqrt-pow1 98.7%

        \[\leadsto -0.5 \cdot \left({x}^{-2.5} + \left(-\color{blue}{{x}^{\left(\frac{-3}{2}\right)}}\right)\right) \]

    19. metadata-eval 98.7%

        \[\leadsto -0.5 \cdot \left({x}^{-2.5} + \left(-{x}^{\color{blue}{-1.5}}\right)\right) \]

11. Applied egg-rr 98.7%

    \[\leadsto -0.5 \cdot \color{blue}{\left({x}^{-2.5} + \left(-{x}^{-1.5}\right)\right)} \]
12. Step-by-step derivation

    1. sub-neg 98.7%

       \[\leadsto -0.5 \cdot \color{blue}{\left({x}^{-2.5} - {x}^{-1.5}\right)} \]

13. Simplified 98.7%

    \[\leadsto -0.5 \cdot \color{blue}{\left({x}^{-2.5} - {x}^{-1.5}\right)} \]

14. Final simplification 98.7%

    \[\leadsto -0.5 \cdot \left({x}^{-2.5} - {x}^{-1.5}\right) \]
15. Add Preprocessing

Alternative 4: 97.9% 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 39.0%

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

3. Taylor expanded in x around inf 70.3%

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

   1. pow1 70.3%

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

   2. pow-flip 71.2%

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

   3. sqrt-pow1 98.6%

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

   4. metadata-eval 98.6%

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

   5. metadata-eval 98.6%

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

5. Applied egg-rr 98.6%

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

   1. unpow1 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 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 39.0%

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

   1. expm1-log1p-u 39.0%

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

   2. expm1-undefine 4.7%

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

   3. inv-pow 4.7%

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

   4. sqrt-pow2 4.7%

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

   5. metadata-eval 4.7%

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

4. Applied egg-rr 4.7%

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

   1. sub-neg 4.7%

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

   2. log1p-undefine 4.7%

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

   3. rem-exp-log 4.7%

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

   4. +-commutative 4.7%

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

   5. metadata-eval 4.7%

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

   6. associate-+l+ 31.1%

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

   7. metadata-eval 31.1%

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

   8. +-rgt-identity 31.1%

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

6. Simplified 31.1%

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

7. Taylor expanded in x around 0 5.6%

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

   1. unpow1/2 5.6%

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

   2. exp-to-pow 5.6%

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

   3. log-rec 5.6%

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

   4. distribute-lft-neg-out 5.6%

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

   5. distribute-rgt-neg-in 5.6%

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

   6. metadata-eval 5.6%

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

   7. exp-to-pow 5.6%

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

9. Simplified 5.6%

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

10. Final simplification 5.6%

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

Developer target: 98.1% 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 2024080 
(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)))))