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

Percentage Accurate: 100.0% → 99.9%

Time: 27.1s

Alternatives: 9

Speedup: 3.5×

• 99.5% 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: 99.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(3 \cdot \log \left(\sqrt[3]{e^{\frac{0.75}{{x}^{4}}}}\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (* (fabs x) (sqrt PI)))
  (+
   (* 3.0 (log (cbrt (exp (/ 0.75 (pow x 4.0))))))
   (+ (/ 0.5 (* x x)) (+ 1.0 (/ 1.875 (pow x 6.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / (fabs(x) * sqrt(((double) M_PI)))) * ((3.0 * log(cbrt(exp((0.75 / pow(x, 4.0)))))) + ((0.5 / (x * x)) + (1.0 + (1.875 / pow(x, 6.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (Math.abs(x) * Math.sqrt(Math.PI))) * ((3.0 * Math.log(Math.cbrt(Math.exp((0.75 / Math.pow(x, 4.0)))))) + ((0.5 / (x * x)) + (1.0 + (1.875 / Math.pow(x, 6.0)))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(abs(x) * sqrt(pi))) * Float64(Float64(3.0 * log(cbrt(exp(Float64(0.75 / (x ^ 4.0)))))) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.0 + Float64(1.875 / (x ^ 6.0))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 * N[Log[N[Power[N[Exp[N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right)} \]
4. Step-by-step derivation

   1. log1p-expm1-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. log1p-udef 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(\color{blue}{0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{4}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(0.75 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-4\right)}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-4\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-4\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {\color{blue}{x}}^{\left(-4\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {x}^{\color{blue}{-4}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\log \color{blue}{\left(\left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)} \cdot \sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right) \cdot \sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\left(\log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)} \cdot \sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \color{blue}{\left({\left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)}^{2}\right)} + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. add-exp-log 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. log1p-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. log1p-expm1-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{e^{\color{blue}{0.75 \cdot {x}^{-4}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{\color{blue}{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-exp-log 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)\right)}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. log1p-def 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.75 \cdot {x}^{-4}\right)\right)}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   10. log1p-expm1-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{0.75 \cdot {x}^{-4}}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   11. exp-prod 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\log \left({\left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\left(\log \left({\left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
8. Step-by-step derivation

   1. log-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

9. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

10. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(3 \cdot \color{blue}{\log \left({\left(e^{\frac{0.75}{{x}^{4}}}\right)}^{0.3333333333333333}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(3 \cdot \log \color{blue}{\left(\sqrt[3]{e^{\frac{0.75}{{x}^{4}}}}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

12. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(3 \cdot \color{blue}{\log \left(\sqrt[3]{e^{\frac{0.75}{{x}^{4}}}}\right)} + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

13. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(3 \cdot \log \left(\sqrt[3]{e^{\frac{0.75}{{x}^{4}}}}\right) + \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (((1.0 / x) + (0.5 / math.pow(x, 3.0))) + (math.pow(x, -5.0) * (0.75 + ((1.875 / x) / x)))) * (math.pow(math.exp(x), x) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (((1.0 / x) + (0.5 / (x ^ 3.0))) + ((x ^ -5.0) * (0.75 + ((1.875 / x) / x)))) * ((exp(x) ^ x) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + \color{blue}{\sqrt{4}}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(\sqrt{4}\right)}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(\sqrt{4}\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. associate-*l* 100.0%

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

   8. associate-*l* 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. fabs-sqr 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. inv-pow 100.0%

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

   13. associate-*r* 100.0%

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

   14. clear-num 100.0%

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

   15. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left({x}^{-1} \cdot {x}^{-4}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{\left(-1 + -4\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. expm1-log1p-u 100.0%

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

   4. expm1-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

    1. unpow1 98.6%

       \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{1}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    2. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    3. fabs-sqr 98.6%

       \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    4. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{1}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    5. unpow1 98.6%

       \[\leadsto \left(\left(\frac{1}{\color{blue}{x}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    6. associate-*r/ 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2} \cdot \left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    7. metadata-eval 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    8. associate-/r* 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    9. unpow1 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{1}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    10. sqr-pow 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    11. fabs-sqr 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    12. sqr-pow 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{1}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    13. unpow1 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    14. associate-/r* 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5}{{x}^{2} \cdot x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    15. unpow2 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{\left(x \cdot x\right)} \cdot x}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    16. unpow3 98.6%

        \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{{x}^{3}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \frac{1.875}{{x}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (+ (+ (/ 1.0 x) (/ 0.5 (pow x 3.0))) (/ 1.875 (pow x 7.0)))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * (((1.0 / x) + (0.5 / pow(x, 3.0))) + (1.875 / pow(x, 7.0)));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * (((1.0 / x) + (0.5 / Math.pow(x, 3.0))) + (1.875 / Math.pow(x, 7.0)));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * (((1.0 / x) + (0.5 / math.pow(x, 3.0))) + (1.875 / math.pow(x, 7.0)))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(Float64(1.0 / x) + Float64(0.5 / (x ^ 3.0))) + Float64(1.875 / (x ^ 7.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * (((1.0 / x) + (0.5 / (x ^ 3.0))) + (1.875 / (x ^ 7.0)));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 98.6%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1.875}{{x}^{7}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around 0 98.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{1}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. sqr-pow 98.6%

      \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. fabs-sqr 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. sqr-pow 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{1}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{x}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. associate-*r/ 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2} \cdot \left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. metadata-eval 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. associate-/r* 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{1}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. fabs-sqr 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{1}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. associate-/r* 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5}{{x}^{2} \cdot x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. unpow2 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{\left(x \cdot x\right)} \cdot x}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. unpow3 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{{x}^{3}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

9. Simplified 98.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

10. Final simplification 98.6%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \frac{1.875}{{x}^{7}}\right) \]

Alternative 4: 99.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (+ (/ 1.0 x) (/ 0.5 (pow x 3.0))) (/ 1.875 (pow x 7.0)))
  (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((1.0 / x) + (0.5 / pow(x, 3.0))) + (1.875 / pow(x, 7.0))) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((1.0 / x) + (0.5 / Math.pow(x, 3.0))) + (1.875 / Math.pow(x, 7.0))) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((1.0 / x) + (0.5 / math.pow(x, 3.0))) + (1.875 / math.pow(x, 7.0))) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 / x) + Float64(0.5 / (x ^ 3.0))) + Float64(1.875 / (x ^ 7.0))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 / x) + (0.5 / (x ^ 3.0))) + (1.875 / (x ^ 7.0))) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 98.6%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1.875}{{x}^{7}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Taylor expanded in x around 0 98.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{1}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. sqr-pow 98.6%

      \[\leadsto \left(\left(\frac{1}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. fabs-sqr 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. sqr-pow 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{{x}^{1}}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{\color{blue}{x}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. associate-*r/ 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2} \cdot \left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. metadata-eval 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{2} \cdot \left|x\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. associate-/r* 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{\frac{0.5}{{x}^{2}}}{\left|x\right|}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. unpow1 98.6%

      \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{1}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. fabs-sqr 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   12. sqr-pow 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{{x}^{1}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   13. unpow1 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{\frac{0.5}{{x}^{2}}}{\color{blue}{x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   14. associate-/r* 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \color{blue}{\frac{0.5}{{x}^{2} \cdot x}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. unpow2 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{\left(x \cdot x\right)} \cdot x}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. unpow3 98.6%

       \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{\color{blue}{{x}^{3}}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

9. Simplified 98.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. pow-exp 98.6%

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

11. Applied egg-rr 98.6%

    \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

12. Final simplification 98.6%

    \[\leadsto \left(\left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \frac{1.875}{{x}^{7}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

Alternative 5: 99.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $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 99.9%

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

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. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + \color{blue}{\sqrt{4}}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(\sqrt{4}\right)}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(\sqrt{4}\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. associate-*l* 100.0%

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

   8. associate-*l* 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. fabs-sqr 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. inv-pow 100.0%

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

   13. associate-*r* 100.0%

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

   14. clear-num 100.0%

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

   15. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left({x}^{-1} \cdot {x}^{-4}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{{x}^{\left(-1 + -4\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. expm1-log1p-u 100.0%

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

   4. expm1-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around inf 98.6%

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

    1. unpow1 98.6%

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

    2. sqr-pow 98.6%

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

    3. fabs-sqr 98.6%

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

    4. sqr-pow 98.6%

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

    5. unpow1 98.6%

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

12. Simplified 98.6%

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

    1. pow-exp 98.6%

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

14. Applied egg-rr 98.6%

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

15. Final simplification 98.6%

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

Alternative 6: 99.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow (exp x) x) (* x (sqrt PI))))

C

double code(double x) {
	return pow(exp(x), x) / (x * sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return math.pow(math.exp(x), x) / (x * math.sqrt(math.pi))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 98.6%

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

7. Taylor expanded in x around inf 98.4%

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

   1. unpow2 98.4%

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

9. Simplified 98.4%

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

    1. associate-*r/ 98.4%

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

    2. add-sqr-sqrt 98.4%

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

    3. fabs-sqr 98.4%

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

    4. add-sqr-sqrt 98.4%

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

    5. associate-*r/ 98.4%

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

    6. sqrt-div 98.4%

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

    7. metadata-eval 98.4%

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

    8. frac-times 98.4%

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

    9. *-un-lft-identity 98.4%

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

    10. pow-exp 98.4%

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \]

Alternative 7: 50.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 98.6%

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

7. Taylor expanded in x around inf 98.4%

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

   1. unpow2 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in x around 0 53.2%

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

    1. unpow2 53.2%

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

12. Simplified 53.2%

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

13. Final simplification 53.2%

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

Alternative 8: 5.4% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 98.6%

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

7. Taylor expanded in x around inf 98.4%

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

   1. unpow2 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in x around 0 53.2%

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

    1. +-commutative 53.2%

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

    2. unpow2 53.2%

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

    3. associate-/l* 5.7%

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

    4. rem-square-sqrt 5.7%

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

    5. fabs-sqr 5.7%

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

    6. rem-square-sqrt 5.7%

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

    7. rem-square-sqrt 5.7%

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

    8. fabs-sqr 5.7%

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

    9. rem-square-sqrt 5.7%

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

12. Simplified 5.7%

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

13. Final simplification 5.7%

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

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 99.9%

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

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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 98.6%

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

7. Taylor expanded in x around inf 98.4%

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

   1. unpow2 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in x around 0 2.5%

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

    1. associate-*r/ 2.5%

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

    2. *-rgt-identity 2.5%

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

    3. rem-square-sqrt 2.5%

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

    4. fabs-sqr 2.5%

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

    5. rem-square-sqrt 2.5%

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

12. Simplified 2.5%

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

13. Final simplification 2.5%

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

Reproduce

herbie shell --seed 2023283 
(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)))))))