2isqrt (example 3.6)

Percentage Accurate: 69.5% → 99.4%

Time: 11.2s

Alternatives: 14

Speedup: 1.8×

Specification

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: 69.5% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. add-log-exp 56.5%

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

   2. *-un-lft-identity 56.5%

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

   3. log-prod 56.5%

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

   4. metadata-eval 56.5%

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

   5. add-log-exp 70.8%

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

   6. inv-pow 70.8%

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

   7. sqrt-pow2 68.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   8. +-commutative 68.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

   9. metadata-eval 68.2%

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

3. Applied egg-rr 68.2%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. +-lft-identity 68.2%

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

5. Simplified 68.2%

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

   1. inv-pow 68.2%

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

   2. sqrt-pow2 71.1%

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

   3. metadata-eval 71.1%

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

   4. metadata-eval 71.1%

      \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

   5. pow-pow 56.3%

      \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

   6. +-commutative 56.3%

      \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

   7. pow1/3 57.3%

      \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

   8. flip-- 57.3%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

7. Applied egg-rr 72.2%

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

   1. *-rgt-identity 72.2%

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

   2. *-rgt-identity 72.2%

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

   3. associate--l+ 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. metadata-eval 91.1%

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

   7. +-inverses 91.1%

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

   8. associate--l+ 72.2%

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

   9. associate--l+ 91.1%

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

   10. associate-*l* 98.9%

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

9. Simplified 98.9%

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

    1. expm1-log1p-u 95.4%

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

    2. expm1-udef 66.5%

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

11. Applied egg-rr 66.5%

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

    1. expm1-def 95.4%

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

    2. expm1-log1p 98.9%

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

    3. associate-/r* 99.4%

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

    4. fma-def 99.4%

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

    5. *-commutative 99.4%

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

    6. fma-def 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 1e-9)
   (/ (+ 1.0 (- x x)) (* x (+ (* (sqrt (/ 1.0 x)) 1.5) (* (sqrt x) 2.0))))
   (- (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)))) <= 1e-9) {
		tmp = (1.0 + (x - x)) / (x * ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0)));
	} 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)))) <= 1d-9) then
        tmp = (1.0d0 + (x - x)) / (x * ((sqrt((1.0d0 / x)) * 1.5d0) + (sqrt(x) * 2.0d0)))
    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)))) <= 1e-9) {
		tmp = (1.0 + (x - x)) / (x * ((Math.sqrt((1.0 / x)) * 1.5) + (Math.sqrt(x) * 2.0)));
	} 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)))) <= 1e-9:
		tmp = (1.0 + (x - x)) / (x * ((math.sqrt((1.0 / x)) * 1.5) + (math.sqrt(x) * 2.0)))
	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)))) <= 1e-9)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(x * Float64(Float64(sqrt(Float64(1.0 / x)) * 1.5) + Float64(sqrt(x) * 2.0))));
	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)))) <= 1e-9)
		tmp = (1.0 + (x - x)) / (x * ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0)));
	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], 1e-9], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.00000000000000006e-9

   1. Initial program 37.2%

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

      1. add-log-exp 6.2%

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

      2. *-un-lft-identity 6.2%

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

      3. log-prod 6.2%

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

      4. metadata-eval 6.2%

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

      5. add-log-exp 37.2%

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

      6. inv-pow 37.2%

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

      7. sqrt-pow2 31.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      8. +-commutative 31.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

      9. metadata-eval 31.4%

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

   3. Applied egg-rr 31.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 31.4%

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

   5. Simplified 31.4%

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

      1. inv-pow 31.4%

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

      2. sqrt-pow2 37.4%

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

      3. metadata-eval 37.4%

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

      4. metadata-eval 37.4%

         \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 5.2%

         \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

      6. +-commutative 5.2%

         \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

      7. pow1/3 7.4%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

      8. flip-- 7.4%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

   7. Applied egg-rr 40.6%

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

      1. *-rgt-identity 40.6%

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

      2. *-rgt-identity 40.6%

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

      3. associate--l+ 81.6%

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

      4. +-inverses 81.6%

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

      5. metadata-eval 81.6%

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

      6. metadata-eval 81.6%

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

      7. +-inverses 81.6%

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

      8. associate--l+ 40.6%

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

      9. associate--l+ 81.6%

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

      10. associate-*l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 98.5%

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

       1. +-commutative 98.5%

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

       2. associate-+r+ 98.5%

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

       3. distribute-rgt-out 98.5%

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

       4. metadata-eval 98.5%

          \[\leadsto \frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} + 2 \cdot \sqrt{x}\right)} \]

   12. Simplified 98.5%

       \[\leadsto \frac{1 + \left(x - x\right)}{x \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot 1.5 + 2 \cdot \sqrt{x}\right)}} \]

   if 1.00000000000000006e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul0-lft 99.9%

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

      5. +-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-9}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. add-log-exp 56.5%

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

   2. *-un-lft-identity 56.5%

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

   3. log-prod 56.5%

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

   4. metadata-eval 56.5%

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

   5. add-log-exp 70.8%

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

   6. inv-pow 70.8%

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

   7. sqrt-pow2 68.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   8. +-commutative 68.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

   9. metadata-eval 68.2%

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

3. Applied egg-rr 68.2%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. +-lft-identity 68.2%

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

5. Simplified 68.2%

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

   1. inv-pow 68.2%

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

   2. sqrt-pow2 71.1%

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

   3. metadata-eval 71.1%

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

   4. metadata-eval 71.1%

      \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

   5. pow-pow 56.3%

      \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

   6. +-commutative 56.3%

      \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

   7. pow1/3 57.3%

      \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

   8. flip-- 57.3%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

7. Applied egg-rr 72.2%

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

   1. *-rgt-identity 72.2%

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

   2. *-rgt-identity 72.2%

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

   3. associate--l+ 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. metadata-eval 91.1%

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

   7. +-inverses 91.1%

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

   8. associate--l+ 72.2%

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

   9. associate--l+ 91.1%

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

   10. associate-*l* 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+16)
   (/ (/ (/ 1.0 x) (+ 1.0 x)) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))
   (/ (+ 1.0 (- x x)) (* x (* (sqrt x) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+16) {
		tmp = ((1.0 / x) / (1.0 + x)) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+16) then
        tmp = ((1.0d0 / x) / (1.0d0 + x)) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    else
        tmp = (1.0d0 + (x - x)) / (x * (sqrt(x) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+16) {
		tmp = ((1.0 / x) / (1.0 + x)) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 + (x - x)) / (x * (Math.sqrt(x) * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+16:
		tmp = ((1.0 / x) / (1.0 + x)) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	else:
		tmp = (1.0 + (x - x)) / (x * (math.sqrt(x) * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+16)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(1.0 + x)) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(x * Float64(sqrt(x) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+16)
		tmp = ((1.0 / x) / (1.0 + x)) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	else
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+16], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e16

   1. Initial program 96.8%

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

      1. flip-- 96.7%

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

      2. div-inv 96.6%

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

      3. frac-times 96.4%

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

      4. metadata-eval 96.4%

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

      5. add-sqr-sqrt 96.6%

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

      6. frac-times 96.6%

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

      7. metadata-eval 96.6%

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

      8. add-sqr-sqrt 96.8%

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

      9. +-commutative 96.8%

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

      10. pow1/2 96.8%

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

      11. pow-flip 96.7%

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

      12. metadata-eval 96.7%

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

      13. inv-pow 96.7%

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

      14. sqrt-pow2 96.7%

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

      15. +-commutative 96.7%

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

      16. metadata-eval 96.7%

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

   3. Applied egg-rr 96.7%

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

      1. associate-*r/ 96.8%

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

      2. *-rgt-identity 96.8%

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

   5. Simplified 96.8%

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

      1. frac-sub 99.3%

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

      2. *-un-lft-identity 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. *-rgt-identity 99.3%

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

      2. associate--l+ 99.3%

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

      3. +-inverses 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-/r* 99.3%

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

   9. Simplified 99.3%

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

   if 1e16 < x

   1. Initial program 36.9%

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

      1. add-log-exp 5.1%

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

      2. *-un-lft-identity 5.1%

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

      3. log-prod 5.1%

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

      4. metadata-eval 5.1%

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

      5. add-log-exp 36.9%

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

      6. inv-pow 36.9%

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

      7. sqrt-pow2 30.6%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      8. +-commutative 30.6%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

      9. metadata-eval 30.6%

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

   3. Applied egg-rr 30.6%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 30.6%

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

   5. Simplified 30.6%

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

      1. inv-pow 30.6%

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

      2. sqrt-pow2 36.9%

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

      3. metadata-eval 36.9%

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

      4. metadata-eval 36.9%

         \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 3.3%

         \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

      6. +-commutative 3.3%

         \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

      7. pow1/3 5.2%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

      8. flip-- 5.2%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

   7. Applied egg-rr 36.9%

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

      1. *-rgt-identity 36.9%

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

      2. *-rgt-identity 36.9%

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

      3. associate--l+ 80.5%

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

      4. +-inverses 80.5%

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

      5. metadata-eval 80.5%

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

      6. metadata-eval 80.5%

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

      7. +-inverses 80.5%

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

      8. associate--l+ 36.9%

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

      9. associate--l+ 80.5%

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

      10. associate-*l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 170000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 170000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (+ 1.0 (- x x)) (* x (* (sqrt x) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 170000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 170000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 + (x - x)) / (x * (sqrt(x) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 170000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 + (x - x)) / (x * (Math.sqrt(x) * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 170000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 + (x - x)) / (x * (math.sqrt(x) * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 170000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(x * Float64(sqrt(x) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 170000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 170000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e8

   1. Initial program 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. clear-num 98.8%

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

      3. associate-/r/ 98.8%

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

      4. prod-diff 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. fma-neg 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. inv-pow 98.8%

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

      9. sqrt-pow2 99.2%

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

      10. metadata-eval 99.2%

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

      11. pow1/2 99.2%

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

      12. pow-flip 99.2%

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

      13. +-commutative 99.2%

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

      14. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.2%

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

      2. distribute-lft1-in 99.2%

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

      3. metadata-eval 99.2%

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

      4. mul0-lft 99.2%

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

      5. +-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   if 1.7e8 < x

   1. Initial program 36.5%

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

      1. add-log-exp 5.2%

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

      2. *-un-lft-identity 5.2%

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

      3. log-prod 5.2%

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

      4. metadata-eval 5.2%

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

      5. add-log-exp 36.5%

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

      6. inv-pow 36.5%

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

      7. sqrt-pow2 30.6%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      8. +-commutative 30.6%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

      9. metadata-eval 30.6%

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

   3. Applied egg-rr 30.6%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 30.6%

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

   5. Simplified 30.6%

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

      1. inv-pow 30.6%

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

      2. sqrt-pow2 36.7%

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

      3. metadata-eval 36.7%

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

      4. metadata-eval 36.7%

         \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 3.8%

         \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

      6. +-commutative 3.8%

         \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

      7. pow1/3 6.0%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

      8. flip-- 6.0%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

   7. Applied egg-rr 39.0%

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

      1. *-rgt-identity 39.0%

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

      2. *-rgt-identity 39.0%

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

      3. associate--l+ 81.1%

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

      4. +-inverses 81.1%

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

      5. metadata-eval 81.1%

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

      6. metadata-eval 81.1%

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

      7. +-inverses 81.1%

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

      8. associate--l+ 39.0%

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

      9. associate--l+ 81.1%

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

      10. associate-*l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 170000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (+ 1.0 (- x x)) (* x (* (+ 1.0 x) (* (sqrt (/ 1.0 x)) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 + (x - x)) / (x * ((1.0 + x) * (sqrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (1.0d0 + (x - x)) / (x * ((1.0d0 + x) * (sqrt((1.0d0 / x)) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 + (x - x)) / (x * ((1.0 + x) * (Math.sqrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (1.0 + (x - x)) / (x * ((1.0 + x) * (math.sqrt((1.0 / x)) * 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(x * Float64(Float64(1.0 + x) * Float64(sqrt(Float64(1.0 / x)) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (1.0 + (x - x)) / (x * ((1.0 + x) * (sqrt((1.0 / x)) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(1.0 + x), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.4%

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

   if 0.680000000000000049 < x

   1. Initial program 37.7%

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

      1. add-log-exp 7.0%

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

      2. *-un-lft-identity 7.0%

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

      3. log-prod 7.0%

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

      4. metadata-eval 7.0%

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

      5. add-log-exp 37.7%

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

      6. inv-pow 37.7%

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

      7. sqrt-pow2 32.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      8. +-commutative 32.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

      9. metadata-eval 32.0%

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

   3. Applied egg-rr 32.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 32.0%

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

   5. Simplified 32.0%

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

      1. inv-pow 32.0%

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

      2. sqrt-pow2 37.9%

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

      3. metadata-eval 37.9%

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

      4. metadata-eval 37.9%

         \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 6.0%

         \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

      6. +-commutative 6.0%

         \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

      7. pow1/3 8.2%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

      8. flip-- 8.2%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

   7. Applied egg-rr 41.1%

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

      1. *-rgt-identity 41.1%

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

      2. *-rgt-identity 41.1%

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

      3. associate--l+ 81.7%

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

      4. +-inverses 81.7%

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

      5. metadata-eval 81.7%

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

      6. metadata-eval 81.7%

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

      7. +-inverses 81.7%

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

      8. associate--l+ 41.1%

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

      9. associate--l+ 81.7%

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

      10. associate-*l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)\right)}\\ \end{array} \]

Alternative 7: 97.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (1.0d0 + (x - x)) / (x * (sqrt(x) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 + (x - x)) / (x * (Math.sqrt(x) * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (1.0 + (x - x)) / (x * (math.sqrt(x) * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(x * Float64(sqrt(x) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (1.0 + (x - x)) / (x * (sqrt(x) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.4%

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

   if 1 < x

   1. Initial program 37.7%

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

      1. add-log-exp 7.0%

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

      2. *-un-lft-identity 7.0%

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

      3. log-prod 7.0%

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

      4. metadata-eval 7.0%

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

      5. add-log-exp 37.7%

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

      6. inv-pow 37.7%

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

      7. sqrt-pow2 32.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      8. +-commutative 32.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(0 + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right) \]

      9. metadata-eval 32.0%

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

   3. Applied egg-rr 32.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(0 + {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 32.0%

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

   5. Simplified 32.0%

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

      1. inv-pow 32.0%

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

      2. sqrt-pow2 37.9%

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

      3. metadata-eval 37.9%

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

      4. metadata-eval 37.9%

         \[\leadsto {x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 6.0%

         \[\leadsto {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

      6. +-commutative 6.0%

         \[\leadsto {x}^{-0.5} - {\left({\color{blue}{\left(x + 1\right)}}^{-1.5}\right)}^{0.3333333333333333} \]

      7. pow1/3 8.2%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

      8. flip-- 8.2%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{-1.5}} \cdot \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}{{x}^{-0.5} + \sqrt[3]{{\left(x + 1\right)}^{-1.5}}}} \]

   7. Applied egg-rr 41.1%

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

      1. *-rgt-identity 41.1%

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

      2. *-rgt-identity 41.1%

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

      3. associate--l+ 81.7%

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

      4. +-inverses 81.7%

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

      5. metadata-eval 81.7%

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

      6. metadata-eval 81.7%

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

      7. +-inverses 81.7%

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

      8. associate--l+ 41.1%

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

      9. associate--l+ 81.7%

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

      10. associate-*l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 96.6%

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

4. Final simplification 97.6%

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

Alternative 8: 69.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.4%

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

   if 1 < x

   1. Initial program 37.7%

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

      1. flip-- 37.7%

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

      2. div-inv 37.7%

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

      3. frac-times 21.5%

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

      4. metadata-eval 21.5%

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

      5. add-sqr-sqrt 22.2%

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

      6. frac-times 27.1%

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

      7. metadata-eval 27.1%

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

      8. add-sqr-sqrt 38.0%

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

      9. +-commutative 38.0%

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

      10. pow1/2 38.0%

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

      11. pow-flip 38.0%

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

      12. metadata-eval 38.0%

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

      13. inv-pow 38.0%

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

      14. sqrt-pow2 38.0%

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

      15. +-commutative 38.0%

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

      16. metadata-eval 38.0%

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

   3. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.0%

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

      2. *-rgt-identity 38.0%

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

   5. Simplified 38.0%

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

   6. Taylor expanded in x around 0 35.3%

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

   7. Taylor expanded in x around inf 36.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{{x}^{2}}}}{{x}^{-0.5} + 1} \]
   8. Step-by-step derivation

      1. unpow2 36.9%

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

   9. Simplified 36.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}}}{{x}^{-0.5} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

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

Alternative 9: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0) (+ (+ (pow x -0.5) (* x 0.5)) -1.0) (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.4%

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

   if 2 < x

   1. Initial program 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. clear-num 37.7%

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

      3. associate-/r/ 37.7%

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

      4. prod-diff 37.7%

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

      5. *-un-lft-identity 37.7%

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

      6. fma-neg 37.7%

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

      7. *-un-lft-identity 37.7%

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

      8. inv-pow 37.7%

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

      9. sqrt-pow2 31.9%

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

      10. metadata-eval 31.9%

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

      11. pow1/2 31.9%

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

      12. pow-flip 37.9%

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

      13. +-commutative 37.9%

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

      14. metadata-eval 37.9%

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

   3. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 37.9%

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

      2. distribute-lft1-in 37.9%

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

      3. metadata-eval 37.9%

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

      4. mul0-lft 37.9%

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

      5. +-rgt-identity 37.9%

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

   5. Simplified 37.9%

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

      1. add-cbrt-cube 8.3%

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

      2. rem-cube-cbrt 8.2%

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

      3. add-cbrt-cube 8.2%

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

      4. pow-pow 8.2%

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

      5. +-commutative 8.2%

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

      6. metadata-eval 8.2%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{\color{blue}{-1.5}}} \]

   7. Applied egg-rr 8.2%

      \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

   8. Taylor expanded in x around inf 5.7%

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

      1. inv-pow 5.7%

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

      2. sqrt-pow1 5.7%

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

      3. metadata-eval 5.7%

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

      4. metadata-eval 5.7%

         \[\leadsto {x}^{\color{blue}{\left(2 \cdot -0.25\right)}} \]

      5. pow-pow 36.1%

         \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]

      6. pow2 36.1%

         \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{-0.25} \]

   10. Applied egg-rr 36.1%

       \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 10: 53.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.82) (+ (pow x -0.5) -1.0) (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.82d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.82:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.82)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.82], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.9%

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

   if 0.819999999999999951 < x

   1. Initial program 37.7%

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

      1. flip-- 37.7%

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

      2. div-inv 37.7%

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

      3. frac-times 21.5%

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

      4. metadata-eval 21.5%

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

      5. add-sqr-sqrt 22.2%

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

      6. frac-times 27.1%

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

      7. metadata-eval 27.1%

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

      8. add-sqr-sqrt 38.0%

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

      9. +-commutative 38.0%

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

      10. pow1/2 38.0%

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

      11. pow-flip 38.0%

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

      12. metadata-eval 38.0%

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

      13. inv-pow 38.0%

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

      14. sqrt-pow2 38.0%

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

      15. +-commutative 38.0%

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

      16. metadata-eval 38.0%

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

   3. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.0%

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

      2. *-rgt-identity 38.0%

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

   5. Simplified 38.0%

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

   6. Taylor expanded in x around 0 7.9%

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

      1. *-commutative 7.9%

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

      2. +-commutative 7.9%

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

      3. distribute-lft-in 7.9%

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

      4. *-rgt-identity 7.9%

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

   8. Simplified 7.9%

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

   9. Taylor expanded in x around inf 7.9%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 11: 68.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (+ (pow x -0.5) -1.0) (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.8d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.8:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.8)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.9%

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

   if 0.80000000000000004 < x

   1. Initial program 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. clear-num 37.7%

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

      3. associate-/r/ 37.7%

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

      4. prod-diff 37.7%

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

      5. *-un-lft-identity 37.7%

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

      6. fma-neg 37.7%

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

      7. *-un-lft-identity 37.7%

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

      8. inv-pow 37.7%

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

      9. sqrt-pow2 31.9%

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

      10. metadata-eval 31.9%

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

      11. pow1/2 31.9%

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

      12. pow-flip 37.9%

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

      13. +-commutative 37.9%

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

      14. metadata-eval 37.9%

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

   3. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 37.9%

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

      2. distribute-lft1-in 37.9%

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

      3. metadata-eval 37.9%

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

      4. mul0-lft 37.9%

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

      5. +-rgt-identity 37.9%

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

   5. Simplified 37.9%

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

      1. add-cbrt-cube 8.3%

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

      2. rem-cube-cbrt 8.2%

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

      3. add-cbrt-cube 8.2%

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

      4. pow-pow 8.2%

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

      5. +-commutative 8.2%

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

      6. metadata-eval 8.2%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{\color{blue}{-1.5}}} \]

   7. Applied egg-rr 8.2%

      \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

   8. Taylor expanded in x around inf 5.7%

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

      1. inv-pow 5.7%

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

      2. sqrt-pow1 5.7%

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

      3. metadata-eval 5.7%

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

      4. metadata-eval 5.7%

         \[\leadsto {x}^{\color{blue}{\left(2 \cdot -0.25\right)}} \]

      5. pow-pow 36.1%

         \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]

      6. pow2 36.1%

         \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{-0.25} \]

   10. Applied egg-rr 36.1%

       \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 12: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 70.8%

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

   1. *-un-lft-identity 70.8%

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

   2. clear-num 70.8%

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

   3. associate-/r/ 70.8%

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

   4. prod-diff 70.8%

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

   5. *-un-lft-identity 70.8%

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

   6. fma-neg 70.8%

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

   7. *-un-lft-identity 70.8%

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

   8. inv-pow 70.8%

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

   9. sqrt-pow2 68.3%

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

   10. metadata-eval 68.3%

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

   11. pow1/2 68.3%

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

   12. pow-flip 71.1%

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

   13. +-commutative 71.1%

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

   14. metadata-eval 71.1%

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

3. Applied egg-rr 71.1%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. fma-udef 71.1%

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

   2. distribute-lft1-in 71.1%

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

   3. metadata-eval 71.1%

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

   4. mul0-lft 71.1%

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

   5. +-rgt-identity 71.1%

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

5. Simplified 71.1%

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

   1. add-cbrt-cube 57.3%

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

   2. rem-cube-cbrt 57.3%

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

   3. add-cbrt-cube 57.3%

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

   4. pow-pow 57.3%

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

   5. +-commutative 57.3%

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

   6. metadata-eval 57.3%

      \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{\color{blue}{-1.5}}} \]

7. Applied egg-rr 57.3%

   \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{-1.5}}} \]

8. Taylor expanded in x around inf 53.3%

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

   1. inv-pow 53.3%

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

   2. sqrt-pow1 53.4%

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

   3. metadata-eval 53.4%

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

   4. *-un-lft-identity 53.4%

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

10. Applied egg-rr 53.4%

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

    1. *-lft-identity 53.4%

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

12. Simplified 53.4%

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

13. Final simplification 53.4%

    \[\leadsto {x}^{-0.5} \]

Alternative 13: 7.4% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 x))

C

double code(double x) {
	return 1.0 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / x;
end

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. flip-- 70.8%

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

   2. div-inv 70.7%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 63.5%

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

   6. frac-times 65.8%

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

   7. metadata-eval 65.8%

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

   8. add-sqr-sqrt 70.8%

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

   9. +-commutative 70.8%

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

   10. pow1/2 70.8%

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

   11. pow-flip 70.8%

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

   12. metadata-eval 70.8%

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

   13. inv-pow 70.8%

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

   14. sqrt-pow2 70.8%

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

   15. +-commutative 70.8%

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

   16. metadata-eval 70.8%

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

3. Applied egg-rr 70.8%

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

   1. associate-*r/ 70.8%

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

   2. *-rgt-identity 70.8%

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

5. Simplified 70.8%

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

6. Taylor expanded in x around 0 55.1%

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

   1. *-commutative 55.1%

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

   2. +-commutative 55.1%

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

   3. distribute-lft-in 55.1%

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

   4. *-rgt-identity 55.1%

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

8. Simplified 55.1%

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

9. Taylor expanded in x around inf 7.5%

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

10. Final simplification 7.5%

    \[\leadsto \frac{1}{x} \]

Alternative 14: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 70.8%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 53.4%

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

3. Taylor expanded in x around inf 1.9%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 98.8% 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 2023256 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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