2isqrt (example 3.6)

Percentage Accurate: 38.1% → 99.6%

Time: 9.9s

Alternatives: 5

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

   1. frac-sub 35.5%

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

   2. div-inv 35.5%

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

   3. *-rgt-identity 35.5%

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

   4. *-un-lft-identity 35.5%

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

   5. +-commutative 35.5%

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

   6. metadata-eval 35.5%

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

   7. frac-times 35.5%

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

   8. associate-*l/ 35.5%

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

   9. *-un-lft-identity 35.5%

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

   10. inv-pow 35.5%

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

   11. sqrt-pow2 35.5%

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

   12. +-commutative 35.5%

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

   13. metadata-eval 35.5%

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

4. Applied egg-rr 35.5%

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

   1. associate-*r/ 35.5%

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

   2. *-rgt-identity 35.5%

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

   3. times-frac 35.5%

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

   4. div-sub 35.5%

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

   5. sub-neg 35.5%

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

   6. *-inverses 35.5%

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

   7. metadata-eval 35.5%

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

   8. /-rgt-identity 35.5%

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

6. Simplified 35.5%

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

   1. flip-+ 35.5%

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

   2. sqrt-undiv 35.5%

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

   3. sqrt-undiv 35.5%

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

   4. add-sqr-sqrt 35.6%

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

   5. metadata-eval 35.6%

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

   6. sqrt-undiv 35.6%

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

8. Applied egg-rr 35.6%

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

   1. *-rgt-identity 35.6%

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

   2. associate-*r/ 31.3%

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

   3. *-commutative 31.3%

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

   4. +-commutative 31.3%

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

   5. distribute-rgt-in 31.3%

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

   6. rgt-mult-inverse 35.6%

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

   7. *-lft-identity 35.6%

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

   8. rem-exp-log 35.6%

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

   9. log1p-undefine 35.6%

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

   10. expm1-define 99.6%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   11. sub-neg 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} + \left(--1\right)}} \cdot {\left(1 + x\right)}^{-0.5} \]

   12. *-lft-identity 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 \cdot \sqrt{\frac{1 + x}{x}}} + \left(--1\right)} \cdot {\left(1 + x\right)}^{-0.5} \]

   13. metadata-eval 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 \cdot \sqrt{\frac{1 + x}{x}} + \color{blue}{1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   14. *-commutative 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} \cdot 1} + 1} \cdot {\left(1 + x\right)}^{-0.5} \]

   15. distribute-lft1-in 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\left(\sqrt{\frac{1 + x}{x}} + 1\right) \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   16. distribute-rgt1-in 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 + \sqrt{\frac{1 + x}{x}} \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   17. *-rgt-identity 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \color{blue}{\sqrt{\frac{1 + x}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   18. *-rgt-identity 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\frac{\color{blue}{\left(1 + x\right) \cdot 1}}{x}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   19. associate-*r/ 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\left(1 + x\right) \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   20. *-commutative 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\frac{1}{x} \cdot \left(1 + x\right)}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   21. +-commutative 99.6%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\frac{1}{x} \cdot \color{blue}{\left(x + 1\right)}}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{1 + \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

11. Taylor expanded in x around 0 99.6%

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

Alternative 2: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

   1. frac-sub 35.5%

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

   2. div-inv 35.5%

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

   3. *-rgt-identity 35.5%

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

   4. *-un-lft-identity 35.5%

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

   5. +-commutative 35.5%

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

   6. metadata-eval 35.5%

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

   7. frac-times 35.5%

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

   8. associate-*l/ 35.5%

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

   9. *-un-lft-identity 35.5%

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

   10. inv-pow 35.5%

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

   11. sqrt-pow2 35.5%

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

   12. +-commutative 35.5%

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

   13. metadata-eval 35.5%

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

4. Applied egg-rr 35.5%

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

   1. associate-*r/ 35.5%

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

   2. *-rgt-identity 35.5%

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

   3. times-frac 35.5%

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

   4. div-sub 35.5%

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

   5. sub-neg 35.5%

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

   6. *-inverses 35.5%

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

   7. metadata-eval 35.5%

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

   8. /-rgt-identity 35.5%

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

6. Simplified 35.5%

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

7. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

9. Simplified 99.4%

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

    1. *-commutative 99.4%

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

    2. associate-*r/ 99.5%

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

    3. sub-neg 99.5%

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

    4. distribute-neg-frac 99.5%

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

    5. metadata-eval 99.5%

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

11. Applied egg-rr 99.5%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

   1. frac-sub 35.5%

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

   2. div-inv 35.5%

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

   3. *-rgt-identity 35.5%

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

   4. *-un-lft-identity 35.5%

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

   5. +-commutative 35.5%

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

   6. metadata-eval 35.5%

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

   7. frac-times 35.5%

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

   8. associate-*l/ 35.5%

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

   9. *-un-lft-identity 35.5%

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

   10. inv-pow 35.5%

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

   11. sqrt-pow2 35.5%

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

   12. +-commutative 35.5%

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

   13. metadata-eval 35.5%

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

4. Applied egg-rr 35.5%

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

   1. associate-*r/ 35.5%

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

   2. *-rgt-identity 35.5%

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

   3. times-frac 35.5%

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

   4. div-sub 35.5%

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

   5. sub-neg 35.5%

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

   6. *-inverses 35.5%

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

   7. metadata-eval 35.5%

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

   8. /-rgt-identity 35.5%

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

6. Simplified 35.5%

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

7. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 98.1% 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 35.5%

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

3. Taylor expanded in x around inf 66.8%

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

   1. pow1 66.8%

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

   2. pow-flip 67.4%

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

   3. sqrt-pow1 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. unpow1 99.2%

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

7. Simplified 99.2%

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

Alternative 5: 35.5% 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 35.5%

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

   1. add-exp-log 5.9%

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

   2. log-rec 5.9%

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

   3. pow1/2 5.9%

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

   4. log-pow 5.9%

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

   5. +-commutative 5.9%

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

   6. log1p-define 5.9%

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

4. Applied egg-rr 5.9%

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

5. Taylor expanded in x around inf 4.9%

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

   1. distribute-lft-neg-in 4.9%

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

   2. metadata-eval 4.9%

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

   3. *-commutative 4.9%

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

   4. exp-to-pow 34.2%

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

   5. unpow1/2 34.2%

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

   6. +-inverses 34.2%

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

7. Simplified 34.2%

   \[\leadsto \color{blue}{0} \]
8. 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 2024110 
(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)))))