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

Percentage Accurate: 100.0% → 100.0%

Time: 13.5s

Alternatives: 10

Speedup: 2.4×

• 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: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{fma}\left(0.75, {t_0}^{5}, \mathsf{fma}\left(1.875, {t_0}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (fma
     0.75
     (pow t_0 5.0)
     (fma 1.875 (pow t_0 7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))
    (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return fma(0.75, pow(t_0, 5.0), fma(1.875, pow(t_0, 7.0), ((1.0 + (0.5 / (x * x))) / fabs(x)))) * (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(fma(0.75, (t_0 ^ 5.0), fma(1.875, (t_0 ^ 7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))) * Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.75 * N[Power[t$95$0, 5.0], $MachinePrecision] + N[(1.875 * N[Power[t$95$0, 7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0))
  (fma
   0.75
   (pow (/ 1.0 (fabs x)) 5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	return (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0)) * fma(0.75, pow((1.0 / fabs(x)), 5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)) * fma(0.75, (Float64(1.0 / abs(x)) ^ 5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision], 5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

10. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

Alternative 3: 100.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x)))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)} - 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{\color{blue}{-5}}\right)} - 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left({x}^{-5}\right)} - 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-5}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

    2. expm1-log1p 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

Alternative 4: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (math.pow(math.exp(x), x) / ((1.0 + (x * math.sqrt(math.pi))) + -1.0)) * (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / ((1.0 + (x * sqrt(pi))) + -1.0)) * (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-div 99.6%

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

   3. sqrt-pow1 99.6%

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

   4. metadata-eval 99.6%

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

   5. add-sqr-sqrt 99.6%

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

   6. fabs-sqr 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. sqrt-div 99.6%

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

   9. sqrt-pow1 99.6%

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

   10. metadata-eval 99.6%

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

   11. add-sqr-sqrt 99.6%

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

   12. fabs-sqr 99.6%

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

   13. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-*r/ 99.6%

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

   3. rem-square-sqrt 99.6%

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

   4. associate-/r* 99.6%

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

   5. pow-sqr 99.6%

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

   6. metadata-eval 99.6%

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

7. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. fabs-sqr 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. expm1-log1p-u 99.6%

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

9. Applied egg-rr 99.6%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)}} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
10. Step-by-step derivation

    1. expm1-udef 99.6%

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

    2. log1p-udef 99.6%

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

    3. rem-exp-log 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (math.pow(math.exp(x), x) / (x * math.sqrt(math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * ((exp(x) ^ x) / (x * sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-div 99.6%

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

   3. sqrt-pow1 99.6%

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

   4. metadata-eval 99.6%

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

   5. add-sqr-sqrt 99.6%

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

   6. fabs-sqr 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. sqrt-div 99.6%

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

   9. sqrt-pow1 99.6%

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

   10. metadata-eval 99.6%

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

   11. add-sqr-sqrt 99.6%

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

   12. fabs-sqr 99.6%

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

   13. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-*r/ 99.6%

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

   3. rem-square-sqrt 99.6%

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

   4. associate-/r* 99.6%

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

   5. pow-sqr 99.6%

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

   6. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 99.6%

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

   1. unpow1 99.6%

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

   2. metadata-eval 99.6%

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

   3. pow-sqr 99.6%

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

   4. unpow1/2 99.6%

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

   5. unpow1/2 99.6%

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

   6. fabs-sqr 99.6%

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

   7. unpow1/2 99.6%

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

   8. unpow1/2 99.6%

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

   9. pow-sqr 99.6%

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

   10. metadata-eval 99.6%

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

   11. unpow1 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 6: 2.3% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
  (/ (pow PI -0.5) x)))

C

double code(double x) {
	return (1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) * (pow(((double) M_PI), -0.5) / x);
}

Java

public static double code(double x) {
	return (1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) * (Math.pow(Math.PI, -0.5) / x);
}

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (math.pow(math.pi, -0.5) / x)

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) * Float64((pi ^ -0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * ((pi ^ -0.5) / x);
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-div 99.6%

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

   3. sqrt-pow1 99.6%

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

   4. metadata-eval 99.6%

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

   5. add-sqr-sqrt 99.6%

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

   6. fabs-sqr 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. sqrt-div 99.6%

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

   9. sqrt-pow1 99.6%

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

   10. metadata-eval 99.6%

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

   11. add-sqr-sqrt 99.6%

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

   12. fabs-sqr 99.6%

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

   13. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-*r/ 99.6%

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

   3. rem-square-sqrt 99.6%

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

   4. associate-/r* 99.6%

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

   5. pow-sqr 99.6%

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

   6. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 2.3%

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

   1. expm1-log1p-u 2.3%

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

   2. expm1-udef 1.7%

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

   3. *-commutative 1.7%

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

   4. add-sqr-sqrt 1.7%

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

   5. fabs-sqr 1.7%

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

   6. add-sqr-sqrt 1.7%

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

   7. associate-/r* 1.7%

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

   8. inv-pow 1.7%

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

   9. sqrt-pow2 1.7%

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

   10. metadata-eval 1.7%

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

10. Applied egg-rr 1.7%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)} - 1\right)} \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
11. Step-by-step derivation

    1. expm1-def 2.3%

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

    2. expm1-log1p 2.3%

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

12. Simplified 2.3%

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

13. Final simplification 2.3%

    \[\leadsto \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{{\pi}^{-0.5}}{x} \]

Alternative 7: 2.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return sqrt((1.0 / ((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.sqrt((1.0 / 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.sqrt((1.0 / 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(sqrt(Float64(1.0 / 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 = sqrt((1.0 / pi)) * (((1.0 / x) + (0.5 / (x ^ 3.0))) + (1.875 / (x ^ 7.0)));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / 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 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

9. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 2.3%

       \[\leadsto 0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) + 1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. associate-+r+ 2.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) + 1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

11. Simplified 2.3%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{0.5}{{x}^{3}} + \left(\frac{0.75}{{x}^{5}} + \frac{1}{x}\right)\right) + \frac{1.875}{{x}^{7}}\right)} \]

12. Taylor expanded in x around inf 2.3%

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

    1. associate-*r/ 2.3%

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

    2. metadata-eval 2.3%

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

14. Simplified 2.3%

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

15. Final simplification 2.3%

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

Alternative 8: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

9. Taylor expanded in x around inf 2.3%

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

    1. +-commutative 2.3%

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

    2. associate-*r/ 2.3%

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

    3. metadata-eval 2.3%

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

    4. rem-square-sqrt 2.3%

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

    5. fabs-sqr 2.3%

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

    6. rem-square-sqrt 2.3%

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

    7. rem-square-sqrt 2.3%

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

    8. fabs-sqr 2.3%

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

    9. rem-square-sqrt 2.3%

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

11. Simplified 2.3%

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

12. Final simplification 2.3%

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

Alternative 9: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

9. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 2.3%

       \[\leadsto 0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) + 1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. associate-+r+ 2.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) + 1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

11. Simplified 2.3%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{0.5}{{x}^{3}} + \left(\frac{0.75}{{x}^{5}} + \frac{1}{x}\right)\right) + \frac{1.875}{{x}^{7}}\right)} \]

12. Taylor expanded in x around inf 2.3%

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

13. Final simplification 2.3%

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

Alternative 10: 2.3% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-udef 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. inv-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. pow-pow 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 7\right)}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, e^{\mathsf{log1p}\left({x}^{\color{blue}{-7}}\right)} - 1, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{e^{\mathsf{log1p}\left({x}^{-7}\right)} - 1}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-7}\right)\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. expm1-log1p 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

7. Simplified 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

9. Taylor expanded in x around inf 2.3%

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

    1. associate-*r* 2.3%

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

    2. *-commutative 2.3%

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

    3. distribute-rgt-out 2.3%

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

    4. associate-*r/ 2.3%

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

    5. metadata-eval 2.3%

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

    6. rem-square-sqrt 2.3%

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

    7. fabs-sqr 2.3%

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

    8. rem-square-sqrt 2.3%

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

    9. pow-plus 2.3%

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

    10. metadata-eval 2.3%

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

    11. +-commutative 2.3%

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

11. Simplified 2.3%

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

12. Taylor expanded in x around inf 2.3%

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

    1. associate-*l/ 2.3%

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

    2. *-lft-identity 2.3%

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

14. Simplified 2.3%

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

15. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2023299 
(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)))))))