2isqrt (example 3.6)

Percentage Accurate: 37.7% → 98.7%

Time: 8.6s

Alternatives: 4

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: 37.7% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.375}{1 + \left(1 + {x}^{1.5}\right)} + 0.5 \cdot {x}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ -0.375 (+ 1.0 (+ 1.0 (pow x 1.5)))) (* 0.5 (pow x -0.5))) x))

C

double code(double x) {
	return ((-0.375 / (1.0 + (1.0 + pow(x, 1.5)))) + (0.5 * pow(x, -0.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.375d0) / (1.0d0 + (1.0d0 + (x ** 1.5d0)))) + (0.5d0 * (x ** (-0.5d0)))) / x
end function

Java

public static double code(double x) {
	return ((-0.375 / (1.0 + (1.0 + Math.pow(x, 1.5)))) + (0.5 * Math.pow(x, -0.5))) / x;
}

Python

def code(x):
	return ((-0.375 / (1.0 + (1.0 + math.pow(x, 1.5)))) + (0.5 * math.pow(x, -0.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-0.375 / Float64(1.0 + Float64(1.0 + (x ^ 1.5)))) + Float64(0.5 * (x ^ -0.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-0.375 / (1.0 + (1.0 + (x ^ 1.5)))) + (0.5 * (x ^ -0.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-0.375 / N[(1.0 + N[(1.0 + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.0%

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

3. Taylor expanded in x around inf 80.3%

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

5. Simplified 80.3%

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

6. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. associate-+r+ 98.4%

      \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.125 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + 0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]

   2. distribute-rgt-out 98.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \left(-0.5 + 0.125\right)} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

   3. metadata-eval 98.4%

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

   4. unpow1/2 98.4%

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

   5. rem-exp-log 94.2%

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

   6. exp-neg 94.2%

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

   7. exp-prod 94.2%

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

   8. distribute-lft-neg-out 94.2%

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

   9. distribute-rgt-neg-in 94.2%

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

   10. metadata-eval 94.2%

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

   11. exp-to-pow 98.4%

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

8. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot -0.375 + 0.5 \cdot {x}^{-0.5}}{x}} \]
9. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. sqrt-div 98.4%

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

   3. metadata-eval 98.4%

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

   4. un-div-inv 98.4%

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

   5. sqrt-pow1 98.4%

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

   6. metadata-eval 98.4%

      \[\leadsto \frac{\frac{-0.375}{{x}^{\color{blue}{1.5}}} + 0.5 \cdot {x}^{-0.5}}{x} \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\color{blue}{\frac{-0.375}{{x}^{1.5}}} + 0.5 \cdot {x}^{-0.5}}{x} \]

12. Applied egg-rr 98.4%

    \[\leadsto \frac{\frac{-0.375}{\color{blue}{e^{\mathsf{log1p}\left({x}^{1.5}\right)} - -1}} + 0.5 \cdot {x}^{-0.5}}{x} \]
13. Step-by-step derivation

    1. sub-neg 98.4%

       \[\leadsto \frac{\frac{-0.375}{\color{blue}{e^{\mathsf{log1p}\left({x}^{1.5}\right)} + \left(--1\right)}} + 0.5 \cdot {x}^{-0.5}}{x} \]

    2. log1p-undefine 98.4%

       \[\leadsto \frac{\frac{-0.375}{e^{\color{blue}{\log \left(1 + {x}^{1.5}\right)}} + \left(--1\right)} + 0.5 \cdot {x}^{-0.5}}{x} \]

    3. rem-exp-log 98.4%

       \[\leadsto \frac{\frac{-0.375}{\color{blue}{\left(1 + {x}^{1.5}\right)} + \left(--1\right)} + 0.5 \cdot {x}^{-0.5}}{x} \]

    4. metadata-eval 98.4%

       \[\leadsto \frac{\frac{-0.375}{\left(1 + {x}^{1.5}\right) + \color{blue}{1}} + 0.5 \cdot {x}^{-0.5}}{x} \]

14. Simplified 98.4%

    \[\leadsto \frac{\frac{-0.375}{\color{blue}{\left(1 + {x}^{1.5}\right) + 1}} + 0.5 \cdot {x}^{-0.5}}{x} \]

15. Final simplification 98.4%

    \[\leadsto \frac{\frac{-0.375}{1 + \left(1 + {x}^{1.5}\right)} + 0.5 \cdot {x}^{-0.5}}{x} \]
16. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-0.5} + \frac{-0.375}{{x}^{1.5}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* 0.5 (pow x -0.5)) (/ -0.375 (pow x 1.5))) x))

C

double code(double x) {
	return ((0.5 * pow(x, -0.5)) + (-0.375 / pow(x, 1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 * (x ** (-0.5d0))) + ((-0.375d0) / (x ** 1.5d0))) / x
end function

Java

public static double code(double x) {
	return ((0.5 * Math.pow(x, -0.5)) + (-0.375 / Math.pow(x, 1.5))) / x;
}

Python

def code(x):
	return ((0.5 * math.pow(x, -0.5)) + (-0.375 / math.pow(x, 1.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 * (x ^ -0.5)) + Float64(-0.375 / (x ^ 1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 * (x ^ -0.5)) + (-0.375 / (x ^ 1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.0%

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

3. Taylor expanded in x around inf 80.3%

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

5. Simplified 80.3%

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

6. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. associate-+r+ 98.4%

      \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.125 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + 0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]

   2. distribute-rgt-out 98.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \left(-0.5 + 0.125\right)} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

   3. metadata-eval 98.4%

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

   4. unpow1/2 98.4%

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

   5. rem-exp-log 94.2%

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

   6. exp-neg 94.2%

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

   7. exp-prod 94.2%

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

   8. distribute-lft-neg-out 94.2%

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

   9. distribute-rgt-neg-in 94.2%

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

   10. metadata-eval 94.2%

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

   11. exp-to-pow 98.4%

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

8. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot -0.375 + 0.5 \cdot {x}^{-0.5}}{x}} \]
9. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. sqrt-div 98.4%

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

   3. metadata-eval 98.4%

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

   4. un-div-inv 98.4%

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

   5. sqrt-pow1 98.4%

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

   6. metadata-eval 98.4%

      \[\leadsto \frac{\frac{-0.375}{{x}^{\color{blue}{1.5}}} + 0.5 \cdot {x}^{-0.5}}{x} \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\color{blue}{\frac{-0.375}{{x}^{1.5}}} + 0.5 \cdot {x}^{-0.5}}{x} \]

11. Final simplification 98.4%

    \[\leadsto \frac{0.5 \cdot {x}^{-0.5} + \frac{-0.375}{{x}^{1.5}}}{x} \]
12. Add Preprocessing

Alternative 3: 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 37.0%

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

3. Taylor expanded in x around inf 68.0%

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

   1. pow1 68.0%

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

   2. *-commutative 68.0%

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

   3. pow-flip 68.7%

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

   4. sqrt-pow1 97.4%

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

   5. metadata-eval 97.4%

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

   6. metadata-eval 97.4%

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

5. Applied egg-rr 97.4%

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

   1. unpow1 97.4%

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

   2. *-commutative 97.4%

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

7. Simplified 97.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]
8. Add Preprocessing

Alternative 4: 34.7% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 37.0%

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

   1. frac-2neg 37.0%

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

   2. metadata-eval 37.0%

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

   3. frac-sub 37.2%

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

   4. *-un-lft-identity 37.2%

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

   5. +-commutative 37.2%

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

   6. +-commutative 37.2%

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

4. Applied egg-rr 37.2%

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

5. Taylor expanded in x around inf 33.5%

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

   1. distribute-rgt1-in 33.5%

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

   2. metadata-eval 33.5%

      \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

   3. mul0-lft 33.5%

      \[\leadsto \color{blue}{0} \]

7. Simplified 33.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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