2isqrt (example 3.6)

Percentage Accurate: 39.1% → 99.2%

Time: 14.6s

Alternatives: 7

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: 39.1% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \mathsf{hypot}\left({\left(x + 1\right)}^{-0.25}, {x}^{-0.25}\right)\right)}^{-2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (* (hypot (sqrt x) x) (hypot (pow (+ x 1.0) -0.25) (pow x -0.25))) -2.0))

C

double code(double x) {
	return pow((hypot(sqrt(x), x) * hypot(pow((x + 1.0), -0.25), pow(x, -0.25))), -2.0);
}

Java

public static double code(double x) {
	return Math.pow((Math.hypot(Math.sqrt(x), x) * Math.hypot(Math.pow((x + 1.0), -0.25), Math.pow(x, -0.25))), -2.0);
}

Python

def code(x):
	return math.pow((math.hypot(math.sqrt(x), x) * math.hypot(math.pow((x + 1.0), -0.25), math.pow(x, -0.25))), -2.0)

Julia

function code(x)
	return Float64(hypot(sqrt(x), x) * hypot((Float64(x + 1.0) ^ -0.25), (x ^ -0.25))) ^ -2.0
end

MATLAB

function tmp = code(x)
	tmp = (hypot(sqrt(x), x) * hypot(((x + 1.0) ^ -0.25), (x ^ -0.25))) ^ -2.0;
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] * N[Sqrt[N[Power[N[(x + 1.0), $MachinePrecision], -0.25], $MachinePrecision] ^ 2 + N[Power[x, -0.25], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 38.0%

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

   1. flip-- 37.9%

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

   2. clear-num 37.9%

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

   3. inv-pow 37.9%

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

   4. sqrt-pow2 37.9%

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

   5. metadata-eval 37.9%

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

   6. inv-pow 37.9%

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

   7. sqrt-pow2 37.9%

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

   8. +-commutative 37.9%

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

   9. metadata-eval 37.9%

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

   10. frac-times 21.0%

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

   11. metadata-eval 21.0%

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

   12. add-sqr-sqrt 18.8%

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

   13. frac-times 26.1%

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

   14. metadata-eval 26.1%

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

   15. add-sqr-sqrt 38.1%

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

   16. +-commutative 38.1%

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

4. Applied egg-rr 38.1%

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

   1. frac-sub 40.3%

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

   2. div-inv 40.3%

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

   3. *-un-lft-identity 40.3%

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

   4. *-rgt-identity 40.3%

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

6. Applied egg-rr 40.3%

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

   1. associate-*r/ 40.3%

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

   2. *-rgt-identity 40.3%

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

   3. associate--l+ 81.0%

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

8. Simplified 81.0%

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

   1. add-sqr-sqrt 80.9%

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

10. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left({\left(x + 1\right)}^{-0.25}, {x}^{-0.25}\right) \cdot \mathsf{hypot}\left(\sqrt{x}, x\right)} \cdot \frac{1}{\mathsf{hypot}\left({\left(x + 1\right)}^{-0.25}, {x}^{-0.25}\right) \cdot \mathsf{hypot}\left(\sqrt{x}, x\right)}} \]
11. Step-by-step derivation

    1. unpow-1 99.2%

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

    2. unpow-1 99.2%

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

    3. pow-sqr 99.2%

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

    4. metadata-eval 99.2%

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

    5. *-commutative 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

    \[\leadsto {\left(\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \mathsf{hypot}\left({\left(x + 1\right)}^{-0.25}, {x}^{-0.25}\right)\right)}^{-2} \]
14. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (/ 0.5 x) (/ 0.125 (pow x 2.0))) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return ((0.5 / x) - (0.125 / pow(x, 2.0))) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 / x) - (0.125d0 / (x ** 2.0d0))) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return ((0.5 / x) - (0.125 / Math.pow(x, 2.0))) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return ((0.5 / x) - (0.125 / math.pow(x, 2.0))) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / x) - Float64(0.125 / (x ^ 2.0))) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / x) - (0.125 / (x ^ 2.0))) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.0%

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

   1. frac-sub 38.0%

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

   2. div-inv 38.0%

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

   3. *-un-lft-identity 38.0%

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

   4. +-commutative 38.0%

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

   5. *-rgt-identity 38.0%

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

   6. metadata-eval 38.0%

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

   7. frac-times 38.0%

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

   8. associate-*l/ 38.0%

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

   9. *-un-lft-identity 38.0%

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

   10. pow1/2 38.0%

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

   11. pow-flip 38.0%

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

   12. +-commutative 38.0%

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

   13. metadata-eval 38.0%

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

4. Applied egg-rr 38.0%

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

   1. *-commutative 38.0%

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

   2. /-rgt-identity 38.0%

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

   3. times-frac 38.0%

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

   4. *-commutative 38.0%

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

   5. times-frac 38.0%

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

   6. div-sub 38.0%

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

   7. sub-neg 38.0%

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

   8. *-inverses 38.0%

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

   9. metadata-eval 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 98.8%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. associate-*r/ 98.8%

      \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. metadata-eval 98.8%

      \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.8%

   \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

10. Final simplification 98.8%

    \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot {\left(x + 1\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x} + -0.125 \cdot {x}^{-2}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 0.5 x) (* -0.125 (pow x -2.0))) (sqrt (+ x 1.0))))

C

double code(double x) {
	return ((0.5 / x) + (-0.125 * pow(x, -2.0))) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 / x) + ((-0.125d0) * (x ** (-2.0d0)))) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((0.5 / x) + (-0.125 * Math.pow(x, -2.0))) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((0.5 / x) + (-0.125 * math.pow(x, -2.0))) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / x) + Float64(-0.125 * (x ^ -2.0))) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / x) + (-0.125 * (x ^ -2.0))) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(N[(0.5 / x), $MachinePrecision] + N[(-0.125 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.0%

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

   1. frac-sub 38.0%

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

   2. div-inv 38.0%

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

   3. *-un-lft-identity 38.0%

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

   4. +-commutative 38.0%

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

   5. *-rgt-identity 38.0%

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

   6. metadata-eval 38.0%

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

   7. frac-times 38.0%

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

   8. associate-*l/ 38.0%

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

   9. *-un-lft-identity 38.0%

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

   10. pow1/2 38.0%

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

   11. pow-flip 38.0%

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

   12. +-commutative 38.0%

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

   13. metadata-eval 38.0%

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

4. Applied egg-rr 38.0%

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

   1. *-commutative 38.0%

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

   2. /-rgt-identity 38.0%

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

   3. times-frac 38.0%

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

   4. *-commutative 38.0%

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

   5. times-frac 38.0%

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

   6. div-sub 38.0%

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

   7. sub-neg 38.0%

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

   8. *-inverses 38.0%

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

   9. metadata-eval 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 98.8%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. associate-*r/ 98.8%

      \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. metadata-eval 98.8%

      \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.8%

   \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
10. Step-by-step derivation

    1. /-rgt-identity 98.8%

       \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot \color{blue}{{\left(1 + x\right)}^{-0.5}} \]

    2. metadata-eval 98.8%

       \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

    3. sqrt-pow2 98.6%

       \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot \color{blue}{{\left(\sqrt{1 + x}\right)}^{-1}} \]

    4. +-commutative 98.6%

       \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot {\left(\sqrt{\color{blue}{x + 1}}\right)}^{-1} \]

    5. inv-pow 98.6%

       \[\leadsto \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot \color{blue}{\frac{1}{\sqrt{x + 1}}} \]

    6. un-div-inv 98.7%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{x} - \frac{0.125}{{x}^{2}}}{\sqrt{x + 1}}} \]

    7. div-inv 98.7%

       \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{0.125 \cdot \frac{1}{{x}^{2}}}}{\sqrt{x + 1}} \]

    8. cancel-sign-sub-inv 98.7%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} + \left(-0.125\right) \cdot \frac{1}{{x}^{2}}}}{\sqrt{x + 1}} \]

    9. metadata-eval 98.7%

       \[\leadsto \frac{\frac{0.5}{x} + \color{blue}{-0.125} \cdot \frac{1}{{x}^{2}}}{\sqrt{x + 1}} \]

    10. pow-flip 98.7%

        \[\leadsto \frac{\frac{0.5}{x} + -0.125 \cdot \color{blue}{{x}^{\left(-2\right)}}}{\sqrt{x + 1}} \]

    11. metadata-eval 98.7%

        \[\leadsto \frac{\frac{0.5}{x} + -0.125 \cdot {x}^{\color{blue}{-2}}}{\sqrt{x + 1}} \]

11. Applied egg-rr 98.7%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{x} + -0.125 \cdot {x}^{-2}}{\sqrt{x + 1}}} \]

12. Final simplification 98.7%

    \[\leadsto \frac{\frac{0.5}{x} + -0.125 \cdot {x}^{-2}}{\sqrt{x + 1}} \]
13. Add Preprocessing

Alternative 4: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.0%

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

   1. frac-sub 38.0%

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

   2. div-inv 38.0%

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

   3. *-un-lft-identity 38.0%

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

   4. +-commutative 38.0%

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

   5. *-rgt-identity 38.0%

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

   6. metadata-eval 38.0%

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

   7. frac-times 38.0%

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

   8. associate-*l/ 38.0%

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

   9. *-un-lft-identity 38.0%

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

   10. pow1/2 38.0%

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

   11. pow-flip 38.0%

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

   12. +-commutative 38.0%

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

   13. metadata-eval 38.0%

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

4. Applied egg-rr 38.0%

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

   1. *-commutative 38.0%

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

   2. /-rgt-identity 38.0%

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

   3. times-frac 38.0%

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

   4. *-commutative 38.0%

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

   5. times-frac 38.0%

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

   6. div-sub 38.0%

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

   7. sub-neg 38.0%

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

   8. *-inverses 38.0%

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

   9. metadata-eval 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 97.2%

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

8. Final simplification 97.2%

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

Alternative 5: 97.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return (0.5 / x) / sqrt((x + 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return (0.5 / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return (0.5 / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.0%

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

   1. frac-sub 38.0%

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

   2. div-inv 38.0%

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

   3. *-un-lft-identity 38.0%

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

   4. +-commutative 38.0%

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

   5. *-rgt-identity 38.0%

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

   6. metadata-eval 38.0%

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

   7. frac-times 38.0%

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

   8. associate-*l/ 38.0%

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

   9. *-un-lft-identity 38.0%

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

   10. pow1/2 38.0%

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

   11. pow-flip 38.0%

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

   12. +-commutative 38.0%

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

   13. metadata-eval 38.0%

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

4. Applied egg-rr 38.0%

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

   1. *-commutative 38.0%

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

   2. /-rgt-identity 38.0%

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

   3. times-frac 38.0%

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

   4. *-commutative 38.0%

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

   5. times-frac 38.0%

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

   6. div-sub 38.0%

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

   7. sub-neg 38.0%

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

   8. *-inverses 38.0%

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

   9. metadata-eval 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 37.0%

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

   1. associate-*r/ 37.0%

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

   2. metadata-eval 37.0%

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

9. Simplified 37.0%

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

    1. /-rgt-identity 37.0%

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

    2. clear-num 37.0%

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

    3. /-rgt-identity 37.0%

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

    4. metadata-eval 37.0%

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

    5. sqrt-pow2 37.0%

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

    6. +-commutative 37.0%

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

    7. inv-pow 37.0%

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

    8. un-div-inv 37.0%

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

    9. associate-+l+ 37.0%

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

11. Applied egg-rr 37.0%

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

    1. remove-double-div 37.0%

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

    2. +-commutative 37.0%

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

    3. associate-+r+ 97.2%

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

    4. metadata-eval 97.2%

       \[\leadsto \frac{\color{blue}{0} + \frac{0.5}{x}}{\sqrt{x + 1}} \]

13. Simplified 97.2%

    \[\leadsto \color{blue}{\frac{0 + \frac{0.5}{x}}{\sqrt{x + 1}}} \]

14. Final simplification 97.2%

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

Alternative 6: 38.4% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.4e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 11.7%

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

      1. frac-sub 11.8%

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

      2. div-inv 11.8%

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

      3. *-un-lft-identity 11.8%

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

      4. +-commutative 11.8%

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

      5. *-rgt-identity 11.8%

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

      6. metadata-eval 11.8%

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

      7. frac-times 11.8%

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

      8. associate-*l/ 11.8%

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

      9. *-un-lft-identity 11.8%

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

      10. pow1/2 11.8%

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

      11. pow-flip 11.8%

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

      12. +-commutative 11.8%

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

      13. metadata-eval 11.8%

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

   4. Applied egg-rr 11.8%

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

      1. *-commutative 11.8%

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

      2. /-rgt-identity 11.8%

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

      3. times-frac 11.8%

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

      4. *-commutative 11.8%

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

      5. times-frac 11.8%

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

      6. div-sub 11.7%

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

      7. sub-neg 11.7%

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

      8. *-inverses 11.7%

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

      9. metadata-eval 11.7%

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

   6. Simplified 11.7%

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

   7. Taylor expanded in x around inf 9.7%

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

      1. associate-*r/ 9.7%

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

      2. metadata-eval 9.7%

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

   9. Simplified 9.7%

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

   10. Taylor expanded in x around 0 5.1%

       \[\leadsto \color{blue}{\left(0.1875 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.25} \]

   11. Taylor expanded in x around 0 8.7%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if 6.4000000000000003e153 < x

   1. Initial program 63.4%

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

   3. Taylor expanded in x around inf 63.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.4% 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 38.0%

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

3. Taylor expanded in x around inf 34.3%

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

4. Final simplification 34.3%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 98.2% 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 2024042 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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