2isqrt (example 3.6)

Percentage Accurate: 38.3% → 98.6%

Time: 18.0s

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: 38.3% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around inf 83.6%

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

   1. *-un-lft-identity 83.6%

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

   2. pow1/2 83.6%

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

   3. pow-flip 83.6%

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

   4. pow-pow 83.6%

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

   5. metadata-eval 83.6%

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

   6. metadata-eval 83.6%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.6%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.6%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.6%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. distribute-rgt-out 98.2%

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

   5. metadata-eval 98.2%

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

10. Simplified 98.2%

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

    1. div-sub 98.2%

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

    2. inv-pow 98.2%

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

    3. sqrt-pow1 98.2%

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

    4. metadata-eval 98.2%

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

    5. associate-/l* 98.2%

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

    6. inv-pow 98.2%

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

    7. sqrt-pow1 98.2%

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

    8. metadata-eval 98.2%

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

12. Applied egg-rr 98.2%

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

13. Final simplification 98.2%

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

Alternative 2: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around inf 83.6%

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

   1. *-un-lft-identity 83.6%

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

   2. pow1/2 83.6%

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

   3. pow-flip 83.6%

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

   4. pow-pow 83.6%

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

   5. metadata-eval 83.6%

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

   6. metadata-eval 83.6%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.6%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.6%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.6%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. distribute-rgt-out 98.2%

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

   5. metadata-eval 98.2%

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

10. Simplified 98.2%

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

    1. div-sub 98.2%

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

    2. inv-pow 98.2%

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

    3. sqrt-pow1 98.2%

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

    4. metadata-eval 98.2%

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

    5. associate-/l* 98.2%

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

    6. inv-pow 98.2%

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

    7. sqrt-pow1 98.2%

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

    8. metadata-eval 98.2%

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

12. Applied egg-rr 98.2%

    \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x} - \frac{{x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]
13. Step-by-step derivation

    1. div-sub 98.2%

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

    2. *-commutative 98.2%

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

    3. distribute-lft-out-- 98.2%

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

14. Simplified 98.2%

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

15. Final simplification 98.2%

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

Alternative 3: 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 42.3%

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

3. Taylor expanded in x around inf 67.7%

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

   1. pow1 67.7%

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

   2. pow1/2 67.7%

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

   3. pow-flip 68.5%

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

   4. pow-pow 97.5%

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

   5. metadata-eval 97.5%

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

   6. metadata-eval 97.5%

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

5. Applied egg-rr 97.5%

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

   1. unpow1 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 4: 35.2% 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 42.3%

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

   1. sub-neg 42.3%

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

   2. +-commutative 42.3%

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

   3. add-sqr-sqrt 24.5%

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

   4. distribute-rgt-neg-in 24.5%

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

   5. fma-define 7.7%

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

   6. pow1/2 7.7%

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

   7. pow-flip 7.7%

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

   8. +-commutative 7.7%

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

   9. metadata-eval 7.7%

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

   10. pow1/2 7.7%

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

   11. pow-flip 7.7%

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

   12. +-commutative 7.7%

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

   13. metadata-eval 7.7%

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

   14. inv-pow 7.7%

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

   15. sqrt-pow2 8.0%

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

   16. metadata-eval 8.0%

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

4. Applied egg-rr 8.0%

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

5. Taylor expanded in x around inf 39.0%

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

   1. distribute-rgt1-in 39.0%

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

   2. metadata-eval 39.0%

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

   3. mul0-lft 39.0%

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

7. Simplified 39.0%

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

8. Final simplification 39.0%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 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 2024085 
(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)))))