Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 14.8s

Alternatives: 9

Speedup: 3.5×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} + -1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (+ (exp (log1p (* (pow x -5.0) (+ 0.75 (/ 1.875 (* x x)))))) -1.0))
  (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0))))

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (exp(log1p((pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))) + -1.0)) * (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0));
}

Java

public static double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.exp(Math.log1p((Math.pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))) + -1.0)) * (Math.pow(Math.exp(x), x) / Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0));
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(exp(log1p(Float64((x ^ -5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x)))))) + -1.0)) * Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)))
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)} - 1\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. *-un-lft-identity 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \color{blue}{\left(1 \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)}\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \color{blue}{\left(0.75 + \frac{\frac{1.875}{x}}{x}\right)}\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. associate-/l/ 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \color{blue}{\frac{1.875}{x \cdot x}}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

11. Final simplification 100.0%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} + -1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

Alternative 2: 100.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} + -1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (+ (exp (log1p (* (pow x -5.0) (+ 0.75 (/ 1.875 (* x x)))))) -1.0))
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (exp(log1p((pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))) + -1.0)) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.exp(Math.log1p((Math.pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))) + -1.0)) * (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (math.exp(math.log1p((math.pow(x, -5.0) * (0.75 + (1.875 / (x * x)))))) + -1.0)) * (math.pow(math.exp(x), x) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(exp(log1p(Float64((x ^ -5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x)))))) + -1.0)) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)} - 1\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. *-un-lft-identity 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \color{blue}{\left(1 \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right)}\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \color{blue}{\left(0.75 + \frac{\frac{1.875}{x}}{x}\right)}\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. associate-/l/ 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \color{blue}{\frac{1.875}{x \cdot x}}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

9. Final simplification 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} + -1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

Alternative 3: 100.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0))
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (* (pow x -5.0) (+ 0.75 (/ (/ 1.875 x) x))))))

C

double code(double x) {
	return (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0)) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + (pow(x, -5.0) * (0.75 + ((1.875 / x) / x))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0)) * (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.pow(x, -5.0) * (0.75 + ((1.875 / x) / x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64((x ^ -5.0) * Float64(0.75 + Float64(Float64(1.875 / x) / x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(N[(1.875 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{-5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)} - 1\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \]

Alternative 4: 100.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (* (pow x -5.0) (+ 0.75 (/ (/ 1.875 x) x))))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + (pow(x, -5.0) * (0.75 + ((1.875 / x) / x))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.pow(x, -5.0) * (0.75 + ((1.875 / x) / x))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (math.pow(x, -5.0) * (0.75 + ((1.875 / x) / x))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64((x ^ -5.0) * Float64(0.75 + Float64(Float64(1.875 / x) / x)))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * (((1.0 + (0.5 / (x * x))) / abs(x)) + ((x ^ -5.0) * (0.75 + ((1.875 / x) / x))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(N[(1.875 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Final simplification 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \]

Alternative 5: 99.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)) (/ 0.75 (pow x 5.0)))
  (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (0.75 / pow(x, 5.0))) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (0.75 / Math.pow(x, 5.0))) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(0.75 / (x ^ 5.0))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{0.75}{{x}^{5}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. pow-exp 99.8%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

9. Applied egg-rr 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

10. Final simplification 99.8%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

Alternative 6: 51.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{1 + x \cdot x}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)) (/ 0.75 (pow x 5.0)))
  (/ (+ 1.0 (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (0.75 / pow(x, 5.0))) * ((1.0 + (x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (0.75 / Math.pow(x, 5.0))) * ((1.0 + (x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * ((1.0 + (x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(0.75 / (x ^ 5.0))) * Float64(Float64(1.0 + Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * ((1.0 + (x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{0.75}{{x}^{5}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 54.9%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. unpow2 54.9%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{1 + \color{blue}{x \cdot x}}{\sqrt{\pi}} \]

10. Simplified 54.9%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + x \cdot x}}{\sqrt{\pi}} \]

11. Final simplification 54.9%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{1 + x \cdot x}{\sqrt{\pi}} \]

Alternative 7: 51.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (/ (* x x) (fabs x))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{0.75}{{x}^{5}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 54.9%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. unpow2 54.9%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{1 + \color{blue}{x \cdot x}}{\sqrt{\pi}} \]

10. Simplified 54.9%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + x \cdot x}}{\sqrt{\pi}} \]

11. Taylor expanded in x around inf 54.9%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
12. Step-by-step derivation

    1. *-commutative 54.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|}} \]

    2. unpow2 54.9%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{x \cdot x}}{\left|x\right|} \]

13. Simplified 54.9%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|}} \]

14. Final simplification 54.9%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|} \]

Alternative 8: 5.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{0.75}{{x}^{5}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 54.9%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. unpow2 54.9%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{1 + \color{blue}{x \cdot x}}{\sqrt{\pi}} \]

10. Simplified 54.9%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + x \cdot x}}{\sqrt{\pi}} \]

11. Taylor expanded in x around inf 54.9%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
12. Step-by-step derivation

    1. *-commutative 54.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|}} \]

    2. unpow2 54.9%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{x \cdot x}}{\left|x\right|} \]

    3. associate-/l* 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{x}{\frac{\left|x\right|}{x}}} \]

    4. unpow1 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\left|\color{blue}{{x}^{1}}\right|}{x}} \]

    5. metadata-eval 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|}{x}} \]

    6. pow-sqr 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|}{x}} \]

    7. unpow1/2 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|}{x}} \]

    8. unpow1/2 5.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|}{x}} \]

    9. fabs-sqr 5.5%

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

    10. unpow1/2 5.5%

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

    11. unpow1/2 5.5%

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

    12. pow-sqr 5.5%

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

    13. metadata-eval 5.5%

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

    14. unpow1 5.5%

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

13. Simplified 5.5%

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

14. Final simplification 5.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{x}{x}} \]

Alternative 9: 2.3% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. associate-/r* 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. unpow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. unpow1/2 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. pow-sqr 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{\left({x}^{\left(2 \cdot 0.5\right)}\right)}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\left({x}^{\color{blue}{1}}\right)}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{\color{blue}{x}}^{\left(\frac{5}{2}\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{-5}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf 99.8%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{0.75}{{x}^{5}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 2.2%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

9. Taylor expanded in x around inf 2.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}} \]
10. Step-by-step derivation

    1. associate-*r/ 2.2%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}} \cdot 1}{\left|x\right|}} \]

    2. *-rgt-identity 2.2%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{\left|x\right|} \]

    3. unpow1 2.2%

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|\color{blue}{{x}^{1}}\right|} \]

    4. metadata-eval 2.2%

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|} \]

    5. pow-sqr 2.2%

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|} \]

    6. unpow1/2 2.2%

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|} \]

    7. unpow1/2 2.2%

       \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|} \]

    8. fabs-sqr 2.2%

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

    9. unpow1/2 2.2%

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

    10. unpow1/2 2.2%

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

    11. pow-sqr 2.2%

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

    12. metadata-eval 2.2%

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

    13. unpow1 2.2%

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

11. Simplified 2.2%

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

12. Final simplification 2.2%

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

Reproduce

herbie shell --seed 2023293 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))