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

Percentage Accurate: 100.0% → 100.0%

Time: 11.4s

Alternatives: 13

Speedup: 3.3×

• 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, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.75 \cdot {x}^{-4} + \mathsf{fma}\left(1.875, {x}^{-6}, 1\right)\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\sqrt[3]{\frac{t\_0}{x}} \cdot {\left(\sqrt[3]{t\_0}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* 0.75 (pow x -4.0)) (fma 1.875 (pow x -6.0) 1.0))))
   (*
    (/ (pow (exp x) x) (cbrt (pow PI 1.5)))
    (+
     (* 0.5 (pow x -3.0))
     (/ (* (cbrt (/ t_0 x)) (pow (cbrt t_0) 2.0)) (pow (cbrt x) 2.0))))))

C

double code(double x) {
	double t_0 = (0.75 * pow(x, -4.0)) + fma(1.875, pow(x, -6.0), 1.0);
	return (pow(exp(x), x) / cbrt(pow(((double) M_PI), 1.5))) * ((0.5 * pow(x, -3.0)) + ((cbrt((t_0 / x)) * pow(cbrt(t_0), 2.0)) / pow(cbrt(x), 2.0)));
}

Julia

function code(x)
	t_0 = Float64(Float64(0.75 * (x ^ -4.0)) + fma(1.875, (x ^ -6.0), 1.0))
	return Float64(Float64((exp(x) ^ x) / cbrt((pi ^ 1.5))) * Float64(Float64(0.5 * (x ^ -3.0)) + Float64(Float64(cbrt(Float64(t_0 / x)) * (cbrt(t_0) ^ 2.0)) / (cbrt(x) ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -6.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(t$95$0 / x), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. pow-flip 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l/ 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]
9. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)\right)}\right)}{x}\right) \]

    2. expm1-undefine 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1}\right)}{x}\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1}\right)}{x}\right) \]
13. Step-by-step derivation

    1. add-cube-cbrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\color{blue}{\left(\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)} \cdot \sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}\right) \cdot \sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}}}{x}\right) \]

    2. add-cube-cbrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\left(\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)} \cdot \sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}\right) \cdot \sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right) \]

    3. times-frac 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)} \cdot \sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1\right)}}{\sqrt[3]{x}}}\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{{\left(\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{\frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}}}\right) \]
15. Step-by-step derivation

    1. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\sqrt[3]{\frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}} \cdot \frac{{\left(\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}\right) \]

    2. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{\sqrt[3]{\frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}} \cdot {\left(\sqrt[3]{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}\right) \]

16. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{\sqrt[3]{\frac{0.75 \cdot {x}^{-4} + \mathsf{fma}\left(1.875, {x}^{-6}, 1\right)}{x}} \cdot {\left(\sqrt[3]{0.75 \cdot {x}^{-4} + \mathsf{fma}\left(1.875, {x}^{-6}, 1\right)}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}\right) \]
17. Add Preprocessing

Alternative 2: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\frac{1.875}{{x}^{6}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (cbrt (pow PI 1.5)))
  (+
   (* 0.5 (pow x -3.0))
   (/ (+ (/ 1.875 (pow x 6.0)) (+ 1.0 (/ 0.75 (pow x 4.0)))) x))))

C

double code(double x) {
	return (pow(exp(x), x) / cbrt(pow(((double) M_PI), 1.5))) * ((0.5 * pow(x, -3.0)) + (((1.875 / pow(x, 6.0)) + (1.0 + (0.75 / pow(x, 4.0)))) / x));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.cbrt(Math.pow(Math.PI, 1.5))) * ((0.5 * Math.pow(x, -3.0)) + (((1.875 / Math.pow(x, 6.0)) + (1.0 + (0.75 / Math.pow(x, 4.0)))) / x));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / cbrt((pi ^ 1.5))) * Float64(Float64(0.5 * (x ^ -3.0)) + Float64(Float64(Float64(1.875 / (x ^ 6.0)) + Float64(1.0 + Float64(0.75 / (x ^ 4.0)))) / x)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. pow-flip 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l/ 100.0%

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

   8. add-sqr-sqrt 100.0%

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

   9. fabs-sqr 100.0%

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

   10. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]
9. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)\right)}\right)}{x}\right) \]

    2. expm1-undefine 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1}\right)}{x}\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, \color{blue}{e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-6}\right)} - 1}\right)}{x}\right) \]

13. Taylor expanded in x around inf 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{1 + \left(0.75 \cdot \frac{1}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}}\right) \]
14. Step-by-step derivation

    1. associate-+r+ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\color{blue}{\left(1 + 0.75 \cdot \frac{1}{{x}^{4}}\right) + \frac{1.875}{{x}^{6}}}}{x}\right) \]

    2. +-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\color{blue}{\frac{1.875}{{x}^{6}} + \left(1 + 0.75 \cdot \frac{1}{{x}^{4}}\right)}}{x}\right) \]

    3. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\frac{1.875}{{x}^{6}} + \left(1 + \color{blue}{\frac{0.75 \cdot 1}{{x}^{4}}}\right)}{x}\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \frac{\frac{1.875}{{x}^{6}} + \left(1 + \frac{\color{blue}{0.75}}{{x}^{4}}\right)}{x}\right) \]

15. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(0.5 \cdot {x}^{-3} + \color{blue}{\frac{\frac{1.875}{{x}^{6}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x}}\right) \]
16. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{{x}^{2}}\right) + \left(1 + \frac{1.875}{{x}^{6}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (/
   (+
    (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (pow x 2.0)))
    (+ 1.0 (/ 1.875 (pow x 6.0))))
   x)))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * ((((0.75 / pow(x, 4.0)) + (0.5 / pow(x, 2.0))) + (1.0 + (1.875 / pow(x, 6.0)))) / x);
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI)) * ((((0.75 / Math.pow(x, 4.0)) + (0.5 / Math.pow(x, 2.0))) + (1.0 + (1.875 / Math.pow(x, 6.0)))) / x);
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.sqrt(math.pi)) * ((((0.75 / math.pow(x, 4.0)) + (0.5 / math.pow(x, 2.0))) + (1.0 + (1.875 / math.pow(x, 6.0)))) / x)

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * Float64(Float64(Float64(Float64(0.75 / (x ^ 4.0)) + Float64(0.5 / (x ^ 2.0))) + Float64(1.0 + Float64(1.875 / (x ^ 6.0)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / sqrt(pi)) * ((((0.75 / (x ^ 4.0)) + (0.5 / (x ^ 2.0))) + (1.0 + (1.875 / (x ^ 6.0)))) / x);
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r+ 100.0%

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

   3. associate-+l+ 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. metadata-eval 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\left(0.75 \cdot {x}^{-5} + 1.875 \cdot {x}^{-7}\right) + \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+
   (+ (* 0.75 (pow x -5.0)) (* 1.875 (pow x -7.0)))
   (/ (fma 0.5 (pow x -2.0) 1.0) x))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * (((0.75 * pow(x, -5.0)) + (1.875 * pow(x, -7.0))) + (fma(0.5, pow(x, -2.0), 1.0) / x));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(0.75 * (x ^ -5.0)) + Float64(1.875 * (x ^ -7.0))) + Float64(fma(0.5, (x ^ -2.0), 1.0) / x)))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(0.75 \cdot {x}^{-5} + 1.875 \cdot {x}^{-7}\right) + \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)} \]
7. Add Preprocessing

Alternative 5: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.5}{{x}^{2}}}{{x}^{2}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (exp (pow x 2.0)) (sqrt PI))
  (* x (+ 1.0 (/ (+ -0.5 (/ -0.5 (pow x 2.0))) (pow x 2.0))))))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / sqrt(((double) M_PI))) / (x * (1.0 + ((-0.5 + (-0.5 / pow(x, 2.0))) / pow(x, 2.0))));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) / Math.sqrt(Math.PI)) / (x * (1.0 + ((-0.5 + (-0.5 / Math.pow(x, 2.0))) / Math.pow(x, 2.0))));
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / math.sqrt(math.pi)) / (x * (1.0 + ((-0.5 + (-0.5 / math.pow(x, 2.0))) / math.pow(x, 2.0))))

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) / Float64(x * Float64(1.0 + Float64(Float64(-0.5 + Float64(-0.5 / (x ^ 2.0))) / (x ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) / sqrt(pi)) / (x * (1.0 + ((-0.5 + (-0.5 / (x ^ 2.0))) / (x ^ 2.0))));
end

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

   3. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. clear-num 99.8%

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

    2. un-div-inv 99.8%

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

    3. pow-exp 99.8%

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

    4. unpow2 99.8%

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

    5. div-inv 99.8%

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

    6. fma-define 99.8%

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

    7. pow-flip 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, \color{blue}{{x}^{\left(-2\right)}}, \frac{0.75}{{x}^{4}}\right)}} \]

    8. metadata-eval 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{\color{blue}{-2}}, \frac{0.75}{{x}^{4}}\right)}} \]

    9. div-inv 99.8%

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

    10. pow-flip 99.8%

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

    11. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Taylor expanded in x around inf 99.8%

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

    1. associate-*r/ 99.8%

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

    2. distribute-lft-in 99.8%

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

    3. metadata-eval 99.8%

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

    4. neg-mul-1 99.8%

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

    5. associate-*r/ 99.8%

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

    6. metadata-eval 99.8%

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

    7. distribute-neg-frac 99.8%

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

    8. metadata-eval 99.8%

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

14. Simplified 99.8%

    \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\color{blue}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.5}{{x}^{2}}}{{x}^{2}}\right)}} \]
15. Add Preprocessing

Alternative 6: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{-2}, 0.75 \cdot {x}^{-4}\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (exp (* x x)) (sqrt PI))
  (/ x (+ 1.0 (fma 0.5 (pow x -2.0) (* 0.75 (pow x -4.0)))))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) / (x / (1.0 + fma(0.5, pow(x, -2.0), (0.75 * pow(x, -4.0)))));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) / Float64(x / Float64(1.0 + fma(0.5, (x ^ -2.0), Float64(0.75 * (x ^ -4.0))))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(x / N[(1.0 + N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

   3. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. clear-num 99.8%

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

    2. un-div-inv 99.8%

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

    3. pow-exp 99.8%

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

    4. unpow2 99.8%

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

    5. div-inv 99.8%

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

    6. fma-define 99.8%

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

    7. pow-flip 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, \color{blue}{{x}^{\left(-2\right)}}, \frac{0.75}{{x}^{4}}\right)}} \]

    8. metadata-eval 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{\color{blue}{-2}}, \frac{0.75}{{x}^{4}}\right)}} \]

    9. div-inv 99.8%

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

    10. pow-flip 99.8%

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

    11. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{-2}, 0.75 \cdot {x}^{-4}\right)}}} \]
12. Step-by-step derivation

    1. unpow2 99.8%

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

13. Applied egg-rr 99.8%

    \[\leadsto \frac{\frac{e^{\color{blue}{x \cdot x}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{-2}, 0.75 \cdot {x}^{-4}\right)}} \]
14. Add Preprocessing

Alternative 7: 99.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

   3. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. unpow2 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)}{x} \]
13. Add Preprocessing

Alternative 8: 99.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{x \cdot \left(1 - \frac{0.5}{{x}^{2}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (exp (pow x 2.0)) (sqrt PI)) (* x (- 1.0 (/ 0.5 (pow x 2.0))))))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / sqrt(((double) M_PI))) / (x * (1.0 - (0.5 / pow(x, 2.0))));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) / Math.sqrt(Math.PI)) / (x * (1.0 - (0.5 / Math.pow(x, 2.0))));
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / math.sqrt(math.pi)) / (x * (1.0 - (0.5 / math.pow(x, 2.0))))

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / sqrt(pi)) / Float64(x * Float64(1.0 - Float64(0.5 / (x ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) / sqrt(pi)) / (x * (1.0 - (0.5 / (x ^ 2.0))));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 - N[(0.5 / N[Power[x, 2.0], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

   3. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. clear-num 99.8%

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

    2. un-div-inv 99.8%

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

    3. pow-exp 99.8%

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

    4. unpow2 99.8%

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

    5. div-inv 99.8%

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

    6. fma-define 99.8%

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

    7. pow-flip 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, \color{blue}{{x}^{\left(-2\right)}}, \frac{0.75}{{x}^{4}}\right)}} \]

    8. metadata-eval 99.8%

       \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\frac{x}{1 + \mathsf{fma}\left(0.5, {x}^{\color{blue}{-2}}, \frac{0.75}{{x}^{4}}\right)}} \]

    9. div-inv 99.8%

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

    10. pow-flip 99.8%

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

    11. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Taylor expanded in x around inf 99.8%

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

    1. associate-*r/ 99.8%

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

    2. metadata-eval 99.8%

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

14. Simplified 99.8%

    \[\leadsto \frac{\frac{e^{{x}^{2}}}{\sqrt{\pi}}}{\color{blue}{x \cdot \left(1 - \frac{0.5}{{x}^{2}}\right)}} \]
15. Add Preprocessing

Alternative 9: 99.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.8%

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

   1. associate-*r/ 99.8%

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

   2. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. unpow2 99.8%

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

11. Applied egg-rr 99.8%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \frac{1 + \frac{0.5}{\color{blue}{x \cdot x}}}{x} \]
12. Add Preprocessing

Alternative 10: 99.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{{x}^{2}}}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (exp (pow x 2.0)) x) (sqrt PI)))

C

double code(double x) {
	return (exp(pow(x, 2.0)) / x) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) / x) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / x) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) / x) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) / x) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.7%

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

   1. sqrt-div 99.7%

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

   2. metadata-eval 99.7%

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

   3. un-div-inv 99.7%

      \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{x}}{\sqrt{\pi}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{e^{{x}^{2}}}{x}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 11: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.8%

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

9. Applied egg-rr 99.7%

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

Alternative 12: 51.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(fma(x, x, 1.0) * Float64((pi ^ -0.5) / x))
end

Wolfram

code[x_] := N[(N[(x * x + 1.0), $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 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.7%

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

8. Taylor expanded in x around 0 51.9%

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

   1. distribute-rgt1-in 51.9%

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

   2. +-commutative 51.9%

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

   3. associate-/l* 51.9%

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

   4. +-commutative 51.9%

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

   5. unpow2 51.9%

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

   6. fma-define 51.9%

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

   7. unpow-1 51.9%

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

   8. metadata-eval 51.9%

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

   9. pow-sqr 51.9%

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

   10. rem-sqrt-square 51.9%

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

   11. rem-square-sqrt 51.9%

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

   12. fabs-sqr 51.9%

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

   13. rem-square-sqrt 51.9%

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

10. Simplified 51.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right) \cdot \frac{{\pi}^{-0.5}}{x}} \]
11. Add Preprocessing

Alternative 13: 2.3% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $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%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 5.1%

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

   2. pow-flip 5.1%

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

   3. add-sqr-sqrt 5.1%

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

   4. fabs-sqr 5.1%

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

   5. add-sqr-sqrt 5.1%

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

   6. metadata-eval 5.1%

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

   7. associate-*l/ 5.1%

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

   8. add-sqr-sqrt 5.1%

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

   9. fabs-sqr 5.1%

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

   10. add-sqr-sqrt 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around inf 99.7%

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

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}} \]
9. 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%

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

   3. unpow-1 2.3%

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

   4. metadata-eval 2.3%

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

   5. pow-sqr 2.3%

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

   6. rem-sqrt-square 2.3%

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

   7. rem-square-sqrt 2.3%

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

   8. fabs-sqr 2.3%

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

   9. rem-square-sqrt 2.3%

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

10. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))))))