2isqrt (example 3.6)

Percentage Accurate: 38.0% → 99.0%

Time: 9.1s

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: 38.0% 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, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 - \frac{0.125 + \frac{-0.0625}{x}}{x}}{x} \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (- 0.5 (/ (+ 0.125 (/ -0.0625 x)) x)) x) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return ((0.5 - ((0.125 + (-0.0625 / x)) / x)) / x) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - ((0.125d0 + ((-0.0625d0) / x)) / x)) / x) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return ((0.5 - ((0.125 + (-0.0625 / x)) / x)) / x) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return ((0.5 - ((0.125 + (-0.0625 / x)) / x)) / x) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(Float64(0.125 + Float64(-0.0625 / x)) / x)) / x) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - ((0.125 + (-0.0625 / x)) / x)) / x) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(N[(0.125 + N[(-0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.5%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-rgt-identity 40.6%

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

   4. *-un-lft-identity 40.6%

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

   5. +-commutative 40.6%

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

   6. metadata-eval 40.6%

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

   7. frac-times 40.6%

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

   8. associate-*l/ 40.6%

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

   9. *-un-lft-identity 40.6%

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

   10. inv-pow 40.6%

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

   11. sqrt-pow2 40.6%

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

   12. +-commutative 40.6%

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

   13. metadata-eval 40.6%

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

4. Applied egg-rr 40.6%

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

   1. associate-*r/ 40.6%

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

   2. *-rgt-identity 40.6%

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

   3. times-frac 40.6%

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

   4. div-sub 40.6%

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

   5. sub-neg 40.6%

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

   6. *-inverses 40.6%

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

   7. metadata-eval 40.6%

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

   8. /-rgt-identity 40.6%

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

6. Simplified 40.6%

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

7. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate-*r/ 99.1%

      \[\leadsto \frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. metadata-eval 99.1%

      \[\leadsto \frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \frac{\color{blue}{0.125}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \frac{0.125}{x}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Taylor expanded in x around -inf 99.1%

    \[\leadsto \frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 - 0.0625 \cdot \frac{1}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]
11. Step-by-step derivation

    1. mul-1-neg 99.1%

       \[\leadsto \frac{0.5 + \color{blue}{\left(-\frac{0.125 - 0.0625 \cdot \frac{1}{x}}{x}\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    2. unsub-neg 99.1%

       \[\leadsto \frac{\color{blue}{0.5 - \frac{0.125 - 0.0625 \cdot \frac{1}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    3. sub-neg 99.1%

       \[\leadsto \frac{0.5 - \frac{\color{blue}{0.125 + \left(-0.0625 \cdot \frac{1}{x}\right)}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    4. associate-*r/ 99.1%

       \[\leadsto \frac{0.5 - \frac{0.125 + \left(-\color{blue}{\frac{0.0625 \cdot 1}{x}}\right)}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    5. metadata-eval 99.1%

       \[\leadsto \frac{0.5 - \frac{0.125 + \left(-\frac{\color{blue}{0.0625}}{x}\right)}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    6. distribute-neg-frac 99.1%

       \[\leadsto \frac{0.5 - \frac{0.125 + \color{blue}{\frac{-0.0625}{x}}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    7. metadata-eval 99.1%

       \[\leadsto \frac{0.5 - \frac{0.125 + \frac{\color{blue}{-0.0625}}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

12. Simplified 99.1%

    \[\leadsto \frac{\color{blue}{0.5 - \frac{0.125 + \frac{-0.0625}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

13. Final simplification 99.1%

    \[\leadsto \frac{0.5 - \frac{0.125 + \frac{-0.0625}{x}}{x}}{x} \cdot {\left(x + 1\right)}^{-0.5} \]
14. Add Preprocessing

Alternative 2: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-rgt-identity 40.6%

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

   4. *-un-lft-identity 40.6%

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

   5. +-commutative 40.6%

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

   6. metadata-eval 40.6%

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

   7. frac-times 40.6%

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

   8. associate-*l/ 40.6%

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

   9. *-un-lft-identity 40.6%

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

   10. inv-pow 40.6%

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

   11. sqrt-pow2 40.6%

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

   12. +-commutative 40.6%

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

   13. metadata-eval 40.6%

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

4. Applied egg-rr 40.6%

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

   1. associate-*r/ 40.6%

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

   2. *-rgt-identity 40.6%

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

   3. times-frac 40.6%

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

   4. div-sub 40.6%

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

   5. sub-neg 40.6%

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

   6. *-inverses 40.6%

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

   7. metadata-eval 40.6%

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

   8. /-rgt-identity 40.6%

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

6. Simplified 40.6%

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

7. Taylor expanded in x around inf 99.0%

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

   1. associate-*r/ 99.0%

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

   2. metadata-eval 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \frac{0.5 - \frac{0.125}{x}}{x} \]
11. 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 40.5%

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

   1. expm1-log1p-u 40.5%

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

   2. expm1-undefine 4.9%

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

   3. inv-pow 4.9%

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

   4. sqrt-pow2 4.9%

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

   5. metadata-eval 4.9%

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

4. Applied egg-rr 4.9%

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

   1. log1p-undefine 4.9%

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

   2. rem-exp-log 4.9%

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

   3. +-commutative 4.9%

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

   4. associate--l+ 32.2%

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

   5. metadata-eval 32.2%

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

   6. +-rgt-identity 32.2%

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

6. Simplified 32.2%

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

7. Taylor expanded in x around inf 66.8%

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

   1. *-commutative 66.8%

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

   2. unpow-1 66.8%

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

   3. exp-to-pow 64.4%

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

   4. *-commutative 64.4%

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

   5. exp-prod 65.1%

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

   6. *-commutative 65.1%

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

   7. associate-*r* 65.1%

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

   8. metadata-eval 65.1%

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

   9. *-commutative 65.1%

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

   10. exp-to-pow 67.6%

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

   11. unpow1/2 67.6%

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

   12. exp-to-pow 65.1%

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

   13. *-commutative 65.1%

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

   14. exp-prod 92.5%

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

   15. *-commutative 92.5%

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

   16. associate-*l* 92.5%

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

   17. metadata-eval 92.5%

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

   18. exp-to-pow 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 7.9% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 0.5 x))

C

double code(double x) {
	return 0.5 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 / x;
}

Python

def code(x):
	return 0.5 / x

Julia

function code(x)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / x;
end

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-rgt-identity 40.6%

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

   4. *-un-lft-identity 40.6%

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

   5. +-commutative 40.6%

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

   6. metadata-eval 40.6%

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

   7. frac-times 40.6%

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

   8. associate-*l/ 40.6%

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

   9. *-un-lft-identity 40.6%

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

   10. inv-pow 40.6%

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

   11. sqrt-pow2 40.6%

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

   12. +-commutative 40.6%

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

   13. metadata-eval 40.6%

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

4. Applied egg-rr 40.6%

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

   1. associate-*r/ 40.6%

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

   2. *-rgt-identity 40.6%

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

   3. times-frac 40.6%

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

   4. div-sub 40.6%

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

   5. sub-neg 40.6%

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

   6. *-inverses 40.6%

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

   7. metadata-eval 40.6%

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

   8. /-rgt-identity 40.6%

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

6. Simplified 40.6%

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

7. Taylor expanded in x around inf 98.4%

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

8. Taylor expanded in x around 0 7.6%

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

9. Final simplification 7.6%

   \[\leadsto \frac{0.5}{x} \]
10. Add Preprocessing

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024157 
(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)))))