2isqrt (example 3.6)

Percentage Accurate: 38.8% → 98.9%

Time: 20.3s

Alternatives: 8

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.8% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{{x}^{5}}}\\ t_1 := \sqrt{\frac{1}{x}}\\ \frac{\left(\mathsf{fma}\left(t\_0, -0.25, 0.5 \cdot t\_1\right) - t\_1 \cdot \frac{0.375}{x}\right) + t\_0 \cdot 0.5}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (pow x 5.0)))) (t_1 (sqrt (/ 1.0 x))))
   (/ (+ (- (fma t_0 -0.25 (* 0.5 t_1)) (* t_1 (/ 0.375 x))) (* t_0 0.5)) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / pow(x, 5.0)));
	double t_1 = sqrt((1.0 / x));
	return ((fma(t_0, -0.25, (0.5 * t_1)) - (t_1 * (0.375 / x))) + (t_0 * 0.5)) / x;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / (x ^ 5.0)))
	t_1 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(fma(t_0, -0.25, Float64(0.5 * t_1)) - Float64(t_1 * Float64(0.375 / x))) + Float64(t_0 * 0.5)) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(t$95$0 * -0.25 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]
9. Step-by-step derivation

10. Simplified 98.9%

    \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x}} \]

11. Final simplification 98.9%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x} \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   0.5
   (pow x -2.5)
   (+ (* -0.25 (pow x -2.5)) (* (pow x -0.5) (+ 0.5 (/ -0.375 x)))))
  x))

C

double code(double x) {
	return fma(0.5, pow(x, -2.5), ((-0.25 * pow(x, -2.5)) + (pow(x, -0.5) * (0.5 + (-0.375 / x))))) / x;
}

Julia

function code(x)
	return Float64(fma(0.5, (x ^ -2.5), Float64(Float64(-0.25 * (x ^ -2.5)) + Float64((x ^ -0.5) * Float64(0.5 + Float64(-0.375 / x))))) / x)
end

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -2.5], $MachinePrecision] + N[(N[(-0.25 * N[Power[x, -2.5], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]
9. Step-by-step derivation

10. Simplified 98.9%

    \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 98.9%

       \[\leadsto \color{blue}{1 \cdot \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x}} \]

12. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, \mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}{x}} \]
13. Step-by-step derivation

    1. *-lft-identity 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, \mathsf{fma}\left({x}^{-2.5}, -0.25, {x}^{-0.5} \cdot 0.5\right) - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}{x}} \]

    2. fma-undefine 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, \color{blue}{\left({x}^{-2.5} \cdot -0.25 + {x}^{-0.5} \cdot 0.5\right)} - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}{x} \]

    3. associate--l+ 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, \color{blue}{{x}^{-2.5} \cdot -0.25 + \left({x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}\right)}\right)}{x} \]

    4. *-commutative 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, \color{blue}{-0.25 \cdot {x}^{-2.5}} + \left({x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}\right)\right)}{x} \]

    5. distribute-lft-out-- 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + \color{blue}{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}\right)}{x} \]

    6. sub-neg 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \color{blue}{\left(0.5 + \left(-\frac{0.375}{x}\right)\right)}\right)}{x} \]

    7. distribute-neg-frac 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right)\right)}{x} \]

    8. metadata-eval 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right)\right)}{x} \]

14. Simplified 98.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\right)}{x}} \]

15. Final simplification 98.9%

    \[\leadsto \frac{\mathsf{fma}\left(0.5, {x}^{-2.5}, -0.25 \cdot {x}^{-2.5} + {x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\right)}{x} \]
16. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{0.5 \cdot t\_0 - t\_0 \cdot \frac{0.375}{x}}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x)))) (/ (- (* 0.5 t_0) (* t_0 (/ 0.375 x))) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return ((0.5 * t_0) - (t_0 * (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / x))
    code = ((0.5d0 * t_0) - (t_0 * (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	return ((0.5 * t_0) - (t_0 * (0.375 / x))) / x;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return ((0.5 * t_0) - (t_0 * (0.375 / x))) / x

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(0.5 * t_0) - Float64(t_0 * Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = ((0.5 * t_0) - (t_0 * (0.375 / x))) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.5 * t$95$0), $MachinePrecision] - N[(t$95$0 * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   6. associate-/l* 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]

11. Final simplification 98.8%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]
12. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-0.5} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (* 0.5 (pow x -0.5)) (* (sqrt (/ 1.0 x)) (/ 0.375 x))) x))

C

double code(double x) {
	return ((0.5 * pow(x, -0.5)) - (sqrt((1.0 / x)) * (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 * (x ** (-0.5d0))) - (sqrt((1.0d0 / x)) * (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	return ((0.5 * Math.pow(x, -0.5)) - (Math.sqrt((1.0 / x)) * (0.375 / x))) / x;
}

Python

def code(x):
	return ((0.5 * math.pow(x, -0.5)) - (math.sqrt((1.0 / x)) * (0.375 / x))) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 * (x ^ -0.5)) - Float64(sqrt(Float64(1.0 / x)) * Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 * (x ^ -0.5)) - (sqrt((1.0 / x)) * (0.375 / x))) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   6. associate-/l* 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]
11. Step-by-step derivation

    1. inv-pow 5.5%

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

    2. sqrt-pow1 5.5%

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

    3. metadata-eval 5.5%

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

    4. *-un-lft-identity 5.5%

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

12. Applied egg-rr 98.8%

    \[\leadsto \frac{0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]
13. Step-by-step derivation

    1. *-lft-identity 5.5%

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

14. Simplified 98.8%

    \[\leadsto \frac{0.5 \cdot \color{blue}{{x}^{-0.5}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

15. Final simplification 98.8%

    \[\leadsto \frac{0.5 \cdot {x}^{-0.5} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]
16. Add Preprocessing

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \cdot \frac{0.5 + \frac{-0.375}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -0.5) (/ (+ 0.5 (/ -0.375 x)) x)))

C

double code(double x) {
	return pow(x, -0.5) * ((0.5 + (-0.375 / x)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) * ((0.5d0 + ((-0.375d0) / x)) / x)
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) * ((0.5 + (-0.375 / x)) / x);
}

Python

def code(x):
	return math.pow(x, -0.5) * ((0.5 + (-0.375 / x)) / x)

Julia

function code(x)
	return Float64((x ^ -0.5) * Float64(Float64(0.5 + Float64(-0.375 / x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) * ((0.5 + (-0.375 / x)) / x);
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   6. associate-/l* 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]
11. Step-by-step derivation

    1. div-sub 98.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]

    2. *-commutative 98.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    3. pow1/2 98.8%

       \[\leadsto \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    4. inv-pow 98.8%

       \[\leadsto \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    5. pow-pow 98.8%

       \[\leadsto \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    6. metadata-eval 98.8%

       \[\leadsto \frac{{x}^{\color{blue}{-0.5}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    7. pow1/2 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot \frac{0.375}{x}}{x} \]

    8. inv-pow 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot \frac{0.375}{x}}{x} \]

    9. pow-pow 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot \frac{0.375}{x}}{x} \]

    10. metadata-eval 98.8%

        \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{x}^{\color{blue}{-0.5}} \cdot \frac{0.375}{x}}{x} \]

12. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]
13. Step-by-step derivation

    1. div-sub 98.8%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]

    2. distribute-lft-out-- 98.8%

       \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]

    3. associate-/l* 98.7%

       \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}} \]

    4. sub-neg 98.7%

       \[\leadsto {x}^{-0.5} \cdot \frac{\color{blue}{0.5 + \left(-\frac{0.375}{x}\right)}}{x} \]

    5. distribute-neg-frac 98.7%

       \[\leadsto {x}^{-0.5} \cdot \frac{0.5 + \color{blue}{\frac{-0.375}{x}}}{x} \]

    6. metadata-eval 98.7%

       \[\leadsto {x}^{-0.5} \cdot \frac{0.5 + \frac{\color{blue}{-0.375}}{x}}{x} \]

14. Simplified 98.7%

    \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{0.5 + \frac{-0.375}{x}}{x}} \]

15. Final simplification 98.7%

    \[\leadsto {x}^{-0.5} \cdot \frac{0.5 + \frac{-0.375}{x}}{x} \]
16. Add Preprocessing

Alternative 6: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (pow x -0.5) (- 0.5 (/ 0.375 x))) x))

C

double code(double x) {
	return (pow(x, -0.5) * (0.5 - (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** (-0.5d0)) * (0.5d0 - (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	return (Math.pow(x, -0.5) * (0.5 - (0.375 / x))) / x;
}

Python

def code(x):
	return (math.pow(x, -0.5) * (0.5 - (0.375 / x))) / x

Julia

function code(x)
	return Float64(Float64((x ^ -0.5) * Float64(0.5 - Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ -0.5) * (0.5 - (0.375 / x))) / x;
end

Wolfram

code[x_] := N[(N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   6. associate-/l* 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]
11. Step-by-step derivation

    1. div-sub 98.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]

    2. *-commutative 98.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    3. pow1/2 98.8%

       \[\leadsto \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    4. inv-pow 98.8%

       \[\leadsto \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    5. pow-pow 98.8%

       \[\leadsto \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    6. metadata-eval 98.8%

       \[\leadsto \frac{{x}^{\color{blue}{-0.5}} \cdot 0.5}{x} - \frac{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x} \]

    7. pow1/2 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot \frac{0.375}{x}}{x} \]

    8. inv-pow 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot \frac{0.375}{x}}{x} \]

    9. pow-pow 98.8%

       \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot \frac{0.375}{x}}{x} \]

    10. metadata-eval 98.8%

        \[\leadsto \frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{x}^{\color{blue}{-0.5}} \cdot \frac{0.375}{x}}{x} \]

12. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5}{x} - \frac{{x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]
13. Step-by-step derivation

    1. div-sub 98.8%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]

    2. distribute-lft-out-- 98.8%

       \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]

14. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x}} \]

15. Final simplification 98.8%

    \[\leadsto \frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x} \]
16. Add Preprocessing

Alternative 7: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 83.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. pow1/2 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. pow-flip 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {\color{blue}{\left({x}^{\left(-5\right)}\right)}}^{0.5}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. pow-pow 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot \color{blue}{{x}^{\left(\left(-5\right) \cdot 0.5\right)}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\left(\color{blue}{-5} \cdot 0.5\right)}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. metadata-eval 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 83.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 83.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
9. Step-by-step derivation

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   6. associate-/l* 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}{x}} \]

11. Taylor expanded in x around inf 98.0%

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

12. Final simplification 98.0%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
13. Add Preprocessing

Alternative 8: 5.6% 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 40.0%

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

3. Taylor expanded in x around 0 5.5%

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

   1. inv-pow 5.5%

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

   2. sqrt-pow1 5.5%

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

   3. metadata-eval 5.5%

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

   4. *-un-lft-identity 5.5%

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

5. Applied egg-rr 5.5%

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

   1. *-lft-identity 5.5%

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

7. Simplified 5.5%

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

8. Final simplification 5.5%

   \[\leadsto {x}^{-0.5} \]
9. Add Preprocessing

Developer target: 98.5% 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 2024115 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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