2isqrt (example 3.6)

Percentage Accurate: 39.0% → 99.6%

Time: 8.6s

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: 39.0% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. add-exp-log 41.4%

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

   2. inv-pow 41.4%

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

   3. sqrt-pow2 31.2%

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

   4. metadata-eval 31.2%

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

   5. pow1/2 31.2%

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

   6. pow-flip 41.4%

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

   7. +-commutative 41.4%

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

   8. metadata-eval 41.4%

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

4. Applied egg-rr 41.4%

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

   1. flip-- 41.4%

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

   2. add-exp-log 41.4%

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

   3. fmm-def 8.8%

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

   4. metadata-eval 8.8%

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

   5. sqrt-pow1 8.8%

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

   6. inv-pow 8.8%

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

   7. metadata-eval 8.8%

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

   8. sqrt-pow1 8.8%

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

   9. inv-pow 8.8%

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

   10. pow-prod-up 8.7%

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

   11. metadata-eval 8.7%

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

   12. inv-pow 8.7%

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

   13. fmm-def 23.1%

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

   14. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 42.5%

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

   1. *-rgt-identity 42.5%

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

   2. *-rgt-identity 42.5%

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

   3. associate--l+ 81.0%

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

   4. +-inverses 81.0%

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

   5. metadata-eval 81.0%

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

   6. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. associate-/r* 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. times-frac 99.3%

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

   4. associate-/l/ 99.4%

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

   5. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.6%

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

    2. *-rgt-identity 99.6%

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

    3. distribute-rgt-in 99.6%

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

    4. pow-plus 99.6%

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

    5. metadata-eval 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in x around 0 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-13)
   (* (pow x -1.5) 0.5)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-13) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-13) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-13) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-13:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-13)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-13)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 2.0000000000000001e-13

   1. Initial program 39.7%

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

      1. flip-- 39.7%

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

      2. frac-times 21.5%

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

      3. metadata-eval 21.5%

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

      4. add-sqr-sqrt 16.3%

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

      5. frac-times 23.8%

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

      6. metadata-eval 23.8%

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

      7. add-sqr-sqrt 39.7%

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

      8. +-commutative 39.7%

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

      9. inv-pow 39.7%

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

      10. sqrt-pow2 39.7%

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

      11. metadata-eval 39.7%

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

      12. pow1/2 39.7%

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

      13. pow-flip 39.7%

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

      14. +-commutative 39.7%

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

      15. metadata-eval 39.7%

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

   4. Applied egg-rr 39.7%

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

   5. Taylor expanded in x around inf 66.6%

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

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

      1. *-un-lft-identity 66.6%

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

      2. pow-flip 67.3%

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

      3. sqrt-pow1 99.4%

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

      4. metadata-eval 99.4%

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

      5. metadata-eval 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. *-lft-identity 99.4%

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

   11. Simplified 99.4%

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

   if 2.0000000000000001e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 86.9%

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

      1. sub-neg 86.9%

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

      2. inv-pow 86.9%

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

      3. sqrt-pow2 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. metadata-eval 87.6%

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

      7. +-commutative 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. *-rgt-identity 87.6%

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

      2. cancel-sign-sub 87.6%

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

      3. distribute-lft-neg-in 87.6%

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

      4. *-rgt-identity 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. metadata-eval 87.6%

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

      7. unpow1/2 87.6%

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

      8. exp-to-pow 86.7%

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

      9. log1p-undefine 86.7%

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

      10. *-commutative 86.7%

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

      11. exp-neg 85.4%

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

      12. *-commutative 85.4%

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

      13. distribute-rgt-neg-in 85.4%

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

      14. log1p-undefine 85.4%

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

      15. metadata-eval 85.4%

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

      16. exp-to-pow 88.0%

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

   6. Simplified 88.0%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. add-exp-log 41.4%

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

   2. inv-pow 41.4%

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

   3. sqrt-pow2 31.2%

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

   4. metadata-eval 31.2%

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

   5. pow1/2 31.2%

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

   6. pow-flip 41.4%

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

   7. +-commutative 41.4%

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

   8. metadata-eval 41.4%

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

4. Applied egg-rr 41.4%

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

   1. flip-- 41.4%

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

   2. add-exp-log 41.4%

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

   3. fmm-def 8.8%

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

   4. metadata-eval 8.8%

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

   5. sqrt-pow1 8.8%

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

   6. inv-pow 8.8%

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

   7. metadata-eval 8.8%

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

   8. sqrt-pow1 8.8%

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

   9. inv-pow 8.8%

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

   10. pow-prod-up 8.7%

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

   11. metadata-eval 8.7%

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

   12. inv-pow 8.7%

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

   13. fmm-def 23.1%

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

   14. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 42.5%

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

   1. *-rgt-identity 42.5%

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

   2. *-rgt-identity 42.5%

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

   3. associate--l+ 81.0%

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

   4. +-inverses 81.0%

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

   5. metadata-eval 81.0%

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

   6. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. distribute-lft-in 98.8%

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

   2. pow1 98.8%

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

   3. pow-prod-up 98.8%

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

   4. metadata-eval 98.8%

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

   5. pow1/2 98.8%

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

10. Applied egg-rr 98.8%

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

Alternative 4: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. flip-- 41.4%

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

   2. frac-times 23.9%

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

   3. metadata-eval 23.9%

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

   4. add-sqr-sqrt 18.8%

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

   5. frac-times 26.1%

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

   6. metadata-eval 26.1%

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

   7. add-sqr-sqrt 41.5%

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

   8. +-commutative 41.5%

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

   9. inv-pow 41.5%

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

   10. sqrt-pow2 41.5%

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

   11. metadata-eval 41.5%

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

   12. pow1/2 41.5%

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

   13. pow-flip 41.5%

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

   14. +-commutative 41.5%

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

   15. metadata-eval 41.5%

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

4. Applied egg-rr 41.5%

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

5. Taylor expanded in x around inf 65.4%

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

   1. *-commutative 65.4%

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

7. Simplified 65.4%

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

   1. *-un-lft-identity 65.4%

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

   2. pow-flip 66.1%

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

   3. sqrt-pow1 97.0%

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

   4. metadata-eval 97.0%

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

   5. metadata-eval 97.0%

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

9. Applied egg-rr 97.0%

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

    1. *-lft-identity 97.0%

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

11. Simplified 97.0%

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

Alternative 5: 36.2% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 41.4%

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

   1. add-sqr-sqrt 23.9%

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

   2. fmm-def 8.1%

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

   3. inv-pow 8.1%

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

   4. sqrt-pow1 8.1%

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

   5. metadata-eval 8.1%

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

   6. inv-pow 8.1%

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

   7. sqrt-pow1 8.1%

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

   8. metadata-eval 8.1%

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

   9. distribute-neg-frac 8.1%

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

   10. metadata-eval 8.1%

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

   11. +-commutative 8.1%

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

4. Applied egg-rr 8.1%

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

5. Taylor expanded in x around inf 37.6%

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

   1. distribute-rgt1-in 37.6%

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

   2. metadata-eval 37.6%

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

   3. mul0-lft 37.6%

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

7. Simplified 37.6%

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

Developer Target 2: 39.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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