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

Percentage Accurate: 100.0% → 100.0%

Time: 26.3s

Alternatives: 13

Speedup: 2.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{0.75}}\\ t_1 := \frac{1}{\left|x\right|}\\ t_2 := {t_1}^{3}\\ \frac{{\left(e^{x}\right)}^{x}}{t_0 \cdot t_0} \cdot \mathsf{fma}\left(1.875, t_2 \cdot \frac{t_2}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} + -1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), t_1\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 0.75))) (t_1 (/ 1.0 (fabs x))) (t_2 (pow t_1 3.0)))
   (*
    (/ (pow (exp x) x) (* t_0 t_0))
    (fma
     1.875
     (* t_2 (/ t_2 (fabs x)))
     (fma
      0.75
      (+ (exp (log1p (* (pow x -3.0) (pow x -2.0)))) -1.0)
      (fma 0.5 (log (exp (pow x -3.0))) t_1))))))

C

double code(double x) {
	double t_0 = cbrt(pow(((double) M_PI), 0.75));
	double t_1 = 1.0 / fabs(x);
	double t_2 = pow(t_1, 3.0);
	return (pow(exp(x), x) / (t_0 * t_0)) * fma(1.875, (t_2 * (t_2 / fabs(x))), fma(0.75, (exp(log1p((pow(x, -3.0) * pow(x, -2.0)))) + -1.0), fma(0.5, log(exp(pow(x, -3.0))), t_1)));
}

Julia

function code(x)
	t_0 = cbrt((pi ^ 0.75))
	t_1 = Float64(1.0 / abs(x))
	t_2 = t_1 ^ 3.0
	return Float64(Float64((exp(x) ^ x) / Float64(t_0 * t_0)) * fma(1.875, Float64(t_2 * Float64(t_2 / abs(x))), fma(0.75, Float64(exp(log1p(Float64((x ^ -3.0) * (x ^ -2.0)))) + -1.0), fma(0.5, log(exp((x ^ -3.0))), t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[Pi, 0.75], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 3.0], $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.875 * N[(t$95$2 * N[(t$95$2 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[(N[Exp[N[Log[1 + N[(N[Power[x, -3.0], $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.5 * N[Log[N[Exp[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + t$95$1), $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%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \color{blue}{e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
7. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \color{blue}{\log \left(e^{{x}^{-3}}\right)}, \frac{1}{\left|x\right|}\right)\right)\right) \]
9. Step-by-step derivation

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

12. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
13. Step-by-step derivation

    1. pow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    2. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left(\sqrt{{\pi}^{1.5}} \cdot \sqrt{{\pi}^{1.5}}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    3. unpow-prod-down 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    4. sqrt-pow1 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.75}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    6. sqrt-pow1 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\pi}^{\color{blue}{0.75}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\pi}^{0.75}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
15. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{0.75}}} \cdot {\left({\pi}^{0.75}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} - 1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]

    2. unpow1/3 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{0.75}} \cdot \sqrt[3]{{\pi}^{0.75}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, e^{\mathsf{log1p}\left({x}^{-3} \cdot {x}^{-2}\right)} + -1, \mathsf{fma}\left(0.5, \log \left(e^{{x}^{-3}}\right), \frac{1}{\left|x\right|}\right)\right)\right) \]
18. Add Preprocessing

Alternative 2: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ {\left(e^{x}\right)}^{x} \cdot \left(t_0 \cdot \left(\frac{1.875}{{x}^{7}} + \left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} + -1\right)\right) + t_0 \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (*
    (pow (exp x) x)
    (+
     (*
      t_0
      (+ (/ 1.875 (pow x 7.0)) (+ (exp (log1p (* 0.75 (pow x -5.0)))) -1.0)))
     (* t_0 (+ (/ 1.0 x) (/ 0.5 (pow x 3.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	return pow(exp(x), x) * ((t_0 * ((1.875 / pow(x, 7.0)) + (exp(log1p((0.75 * pow(x, -5.0)))) + -1.0))) + (t_0 * ((1.0 / x) + (0.5 / pow(x, 3.0)))));
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return Math.pow(Math.exp(x), x) * ((t_0 * ((1.875 / Math.pow(x, 7.0)) + (Math.exp(Math.log1p((0.75 * Math.pow(x, -5.0)))) + -1.0))) + (t_0 * ((1.0 / x) + (0.5 / Math.pow(x, 3.0)))));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	return math.pow(math.exp(x), x) * ((t_0 * ((1.875 / math.pow(x, 7.0)) + (math.exp(math.log1p((0.75 * math.pow(x, -5.0)))) + -1.0))) + (t_0 * ((1.0 / x) + (0.5 / math.pow(x, 3.0)))))

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	return Float64((exp(x) ^ x) * Float64(Float64(t_0 * Float64(Float64(1.875 / (x ^ 7.0)) + Float64(exp(log1p(Float64(0.75 * (x ^ -5.0)))) + -1.0))) + Float64(t_0 * Float64(Float64(1.0 / x) + Float64(0.5 / (x ^ 3.0))))))
end

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

11. Simplified 100.0%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)} \]
12. Step-by-step derivation

    1. exp-prod 99.8%

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

13. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]
14. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)\right)}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

    2. expm1-udef 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{{x}^{5}}\right)} - 1\right)}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

    3. div-inv 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \left(e^{\mathsf{log1p}\left(\color{blue}{0.75 \cdot \frac{1}{{x}^{5}}}\right)} - 1\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

    4. pow-flip 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \left(e^{\mathsf{log1p}\left(0.75 \cdot \color{blue}{{x}^{\left(-5\right)}}\right)} - 1\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

    5. metadata-eval 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{\color{blue}{-5}}\right)} - 1\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

15. Applied egg-rr 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} - 1\right)}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]

16. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \left(e^{\mathsf{log1p}\left(0.75 \cdot {x}^{-5}\right)} + -1\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]
17. Add Preprocessing

Alternative 3: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return Math.pow(Math.exp(x), x) * ((t_0 * ((1.0 / x) + (0.5 / Math.pow(x, 3.0)))) + (t_0 * ((1.875 / Math.pow(x, 7.0)) + (0.75 / Math.pow(x, 5.0)))));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	return math.pow(math.exp(x), x) * ((t_0 * ((1.0 / x) + (0.5 / math.pow(x, 3.0)))) + (t_0 * ((1.875 / math.pow(x, 7.0)) + (0.75 / math.pow(x, 5.0)))))

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = (exp(x) ^ x) * ((t_0 * ((1.0 / x) + (0.5 / (x ^ 3.0)))) + (t_0 * ((1.875 / (x ^ 7.0)) + (0.75 / (x ^ 5.0)))));
end

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

11. Simplified 100.0%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)} \]
12. Step-by-step derivation

    1. exp-prod 99.8%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right)\right) \]
15. Add Preprocessing

Alternative 4: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return ((t_0 * ((1.0 / x) + (0.5 / Math.pow(x, 3.0)))) + (t_0 * ((1.875 / Math.pow(x, 7.0)) + (0.75 / Math.pow(x, 5.0))))) * Math.exp((x * x));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	return ((t_0 * ((1.0 / x) + (0.5 / math.pow(x, 3.0)))) + (t_0 * ((1.875 / math.pow(x, 7.0)) + (0.75 / math.pow(x, 5.0))))) * math.exp((x * x))

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = ((t_0 * ((1.0 / x) + (0.5 / (x ^ 3.0)))) + (t_0 * ((1.875 / (x ^ 7.0)) + (0.75 / (x ^ 5.0))))) * exp((x * x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 100.0%

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

    2. associate-+r+ 100.0%

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

    3. associate-+l+ 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

    \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right)\right) \cdot e^{x \cdot x} \]
13. Add Preprocessing

Alternative 5: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around 0 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return Math.pow(Math.exp(x), x) * ((t_0 / x) + (t_0 * ((0.5 / Math.pow(x, 3.0)) + (0.75 / Math.pow(x, 5.0)))));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	return math.pow(math.exp(x), x) * ((t_0 / x) + (t_0 * ((0.5 / math.pow(x, 3.0)) + (0.75 / math.pow(x, 5.0)))))

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	return Float64((exp(x) ^ x) * Float64(Float64(t_0 / x) + Float64(t_0 * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(0.75 / (x ^ 5.0))))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = (exp(x) ^ x) * ((t_0 / x) + (t_0 * ((0.5 / (x ^ 3.0)) + (0.75 / (x ^ 5.0)))));
end

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

11. Simplified 99.8%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\frac{\sqrt{\frac{1}{\pi}}}{x} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right)} \]
12. Step-by-step derivation

    1. exp-prod 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\frac{\sqrt{\frac{1}{\pi}}}{x} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right) \]
15. Add Preprocessing

Alternative 7: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

11. Simplified 99.8%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\frac{\sqrt{\frac{1}{\pi}}}{x} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 99.8%

       \[\leadsto e^{x \cdot x} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\frac{1}{\pi}}}{x}\right)\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right) \]

    2. expm1-udef 30.3%

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

    3. inv-pow 30.3%

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

    4. sqrt-pow1 30.3%

       \[\leadsto e^{x \cdot x} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right)} - 1\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right) \]

    5. metadata-eval 30.3%

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

13. Applied egg-rr 30.3%

    \[\leadsto e^{x \cdot x} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)} - 1\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right) \]
14. Step-by-step derivation

    1. expm1-def 99.8%

       \[\leadsto e^{x \cdot x} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.75}{{x}^{5}}\right)\right) \]

    2. expm1-log1p 99.8%

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

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 8: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-out 99.8%

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

9. Simplified 99.8%

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

    1. exp-prod 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 9: 99.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-rgt-out 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 10: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

    1. associate-*r* 99.8%

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

    2. distribute-rgt-out 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-*r/ 99.8%

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

    5. metadata-eval 99.8%

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

11. Simplified 99.8%

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

    1. exp-prod 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]
15. Add Preprocessing

Alternative 11: 99.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 7.0%

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

6. Applied egg-rr 7.0%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto e^{x \cdot x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

    1. associate-*r* 99.8%

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

    2. distribute-rgt-out 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-*r/ 99.8%

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

    5. metadata-eval 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \cdot e^{x \cdot x} \]
13. Add Preprocessing

Alternative 12: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around inf 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around inf 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 13: 1.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 30.1%

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

   1. associate-*r* 30.1%

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

   2. *-commutative 30.1%

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

   3. associate-*r/ 30.1%

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

   4. metadata-eval 30.1%

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

   5. unpow2 30.1%

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

   6. unpow1 30.1%

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

   7. metadata-eval 30.1%

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

   8. pow-sqr 30.1%

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

   9. unpow1/2 30.1%

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

   10. unpow1/2 30.1%

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

   11. fabs-sqr 30.1%

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

   12. unpow1/2 30.1%

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

   13. unpow1/2 30.1%

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

   14. pow-sqr 30.1%

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

   15. metadata-eval 30.1%

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

   16. unpow1 30.1%

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

   17. unpow3 30.1%

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

7. Simplified 30.1%

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

8. Taylor expanded in x around 0 1.8%

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

   1. associate-*r* 1.8%

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

   2. associate-*r/ 1.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}} \cdot \sqrt{\frac{1}{\pi}} \]

   3. metadata-eval 1.8%

      \[\leadsto \frac{\color{blue}{0.5}}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}} \]

10. Simplified 1.8%

    \[\leadsto \color{blue}{\frac{0.5}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 1.8%

       \[\leadsto \color{blue}{\sqrt{\frac{0.5}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}} \cdot \sqrt{\frac{0.5}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}}} \]

    2. sqrt-unprod 1.8%

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

    3. *-commutative 1.8%

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

    4. *-commutative 1.8%

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

    5. swap-sqr 1.8%

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

    6. add-sqr-sqrt 1.8%

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

    7. clear-num 1.8%

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

    8. associate-/r/ 1.8%

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

    9. pow-flip 1.8%

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

    10. metadata-eval 1.8%

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

    11. *-commutative 1.8%

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

    12. div-inv 1.8%

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

    13. pow-flip 1.8%

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

    14. metadata-eval 1.8%

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

    15. swap-sqr 1.8%

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

12. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left(0.25 \cdot {x}^{-6}\right)}} \]
13. Step-by-step derivation

    1. associate-*r* 1.8%

       \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{\pi} \cdot 0.25\right) \cdot {x}^{-6}}} \]

    2. *-commutative 1.8%

       \[\leadsto \sqrt{\color{blue}{{x}^{-6} \cdot \left(\frac{1}{\pi} \cdot 0.25\right)}} \]

    3. associate-*l/ 1.8%

       \[\leadsto \sqrt{{x}^{-6} \cdot \color{blue}{\frac{1 \cdot 0.25}{\pi}}} \]

    4. metadata-eval 1.8%

       \[\leadsto \sqrt{{x}^{-6} \cdot \frac{\color{blue}{0.25}}{\pi}} \]

14. Simplified 1.8%

    \[\leadsto \color{blue}{\sqrt{{x}^{-6} \cdot \frac{0.25}{\pi}}} \]
15. Step-by-step derivation

    1. *-commutative 1.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25}{\pi} \cdot {x}^{-6}}} \]

    2. sqrt-prod 1.8%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{\pi}} \cdot \sqrt{{x}^{-6}}} \]

    3. sqrt-div 1.8%

       \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{\pi}}} \cdot \sqrt{{x}^{-6}} \]

    4. metadata-eval 1.8%

       \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{\pi}} \cdot \sqrt{{x}^{-6}} \]

    5. sqrt-pow1 1.8%

       \[\leadsto \frac{0.5}{\sqrt{\pi}} \cdot \color{blue}{{x}^{\left(\frac{-6}{2}\right)}} \]

    6. metadata-eval 1.8%

       \[\leadsto \frac{0.5}{\sqrt{\pi}} \cdot {x}^{\color{blue}{-3}} \]

    7. metadata-eval 1.8%

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

    8. pow-flip 1.8%

       \[\leadsto \frac{0.5}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{{x}^{3}}} \]

    9. div-inv 1.8%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{\pi}}}{{x}^{3}}} \]

    10. unpow3 1.8%

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

    11. unpow2 1.8%

        \[\leadsto \frac{\frac{0.5}{\sqrt{\pi}}}{\color{blue}{{x}^{2}} \cdot x} \]

    12. associate-/l/ 1.8%

        \[\leadsto \color{blue}{\frac{\frac{\frac{0.5}{\sqrt{\pi}}}{x}}{{x}^{2}}} \]

    13. *-un-lft-identity 1.8%

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

    14. unpow2 1.8%

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

    15. times-frac 1.8%

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

    16. associate-/l/ 1.8%

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

16. Applied egg-rr 1.8%

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

17. Final simplification 1.8%

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

Reproduce

herbie shell --seed 2024019 
(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)))))))