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

Percentage Accurate: 100.0% → 100.0%

Time: 16.0s

Alternatives: 10

Speedup: 4.0×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := {t\_0}^{3}\\ \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \mathsf{fma}\left(1.875, t\_1 \cdot {x}^{-4}, \mathsf{fma}\left(0.75, \frac{{x}^{-4}}{\left|x\right|}, \mathsf{fma}\left(0.5, t\_1, t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))) (t_1 (pow t_0 3.0)))
   (*
    (/ (pow (exp x) x) (expm1 (log1p (sqrt PI))))
    (fma
     1.875
     (* t_1 (pow x -4.0))
     (fma 0.75 (/ (pow x -4.0) (fabs x)) (fma 0.5 t_1 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = pow(t_0, 3.0);
	return (pow(exp(x), x) / expm1(log1p(sqrt(((double) M_PI))))) * fma(1.875, (t_1 * pow(x, -4.0)), fma(0.75, (pow(x, -4.0) / fabs(x)), fma(0.5, t_1, t_0)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = t_0 ^ 3.0
	return Float64(Float64((exp(x) ^ x) / expm1(log1p(sqrt(pi)))) * fma(1.875, Float64(t_1 * (x ^ -4.0)), fma(0.75, Float64((x ^ -4.0) / abs(x)), fma(0.5, t_1, t_0))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

6. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)} - 1\right)}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. unpow2 100.0%

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

   4. pow-sqr 100.0%

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

   5. metadata-eval 100.0%

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

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{x}^{-4}}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 100.0%

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

    2. expm1-udef 100.0%

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

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {x}^{-4}, \mathsf{fma}\left(0.75, \frac{\color{blue}{e^{\mathsf{log1p}\left({\left({x}^{-2}\right)}^{2}\right)} - 1}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
13. Step-by-step derivation

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

    3. unpow2 100.0%

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

    4. pow-sqr 100.0%

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

    5. metadata-eval 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {x}^{-4}, \mathsf{fma}\left(0.75, \frac{{x}^{-4}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
16. Add Preprocessing

Alternative 2: 100.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \left(\frac{0.75}{{x}^{5}} + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1.875 \cdot {x}^{-7}\right)\right) + \frac{1}{x}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (*
   (sqrt (/ 1.0 PI))
   (+
    (/ 0.5 (pow x 3.0))
    (+
     (/ 0.75 (pow x 5.0))
     (+ (expm1 (log1p (* 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)) + ((0.75 / pow(x, 5.0)) + (expm1(log1p((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)) + ((0.75 / Math.pow(x, 5.0)) + (Math.expm1(Math.log1p((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)) + ((0.75 / math.pow(x, 5.0)) + (math.expm1(math.log1p((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(0.75 / (x ^ 5.0)) + Float64(expm1(log1p(Float64(1.875 * (x ^ -7.0)))) + Float64(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[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(Exp[N[Log[1 + N[(1.875 * N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

   1. exp-prod 100.0%

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

9. Applied egg-rr 100.0%

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

    1. div-inv 100.0%

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

    2. pow-flip 100.0%

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

    3. metadata-eval 100.0%

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

    4. expm1-log1p-u 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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)) + ((0.75 / Math.pow(x, 5.0)) + ((1.0 / x) + (1.875 / Math.pow(x, 7.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

   1. exp-prod 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $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. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 5: 99.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.2%

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

   1. associate-+r+ 99.2%

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

   2. +-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. associate-*r* 99.2%

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

   6. associate-*r* 99.2%

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

   7. distribute-rgt-out 99.2%

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

   8. associate-*r/ 99.2%

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

   9. metadata-eval 99.2%

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

   10. associate-*r/ 99.2%

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

   11. metadata-eval 99.2%

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

10. Simplified 99.2%

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

    1. expm1-log1p-u 99.2%

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

    2. expm1-udef 32.0%

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

    3. div-inv 32.0%

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

    4. div-inv 32.0%

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

    5. inv-pow 32.0%

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

    6. sqrt-pow1 32.0%

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

    7. metadata-eval 32.0%

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

12. Applied egg-rr 32.0%

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

    1. expm1-def 99.2%

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

    2. expm1-log1p 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 6: 99.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $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. Taylor expanded in x around inf 99.1%

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

   1. associate-+r+ 99.1%

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

   2. +-commutative 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

   5. rem-square-sqrt 99.1%

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

   6. fabs-sqr 99.1%

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

   7. rem-square-sqrt 99.1%

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

   8. associate-*r/ 99.1%

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

   9. metadata-eval 99.1%

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

   10. rem-square-sqrt 99.1%

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

   11. fabs-sqr 99.1%

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

   12. rem-square-sqrt 99.1%

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

   13. rem-square-sqrt 99.1%

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

   14. fabs-sqr 99.1%

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

   15. rem-square-sqrt 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 7: 99.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * (sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + (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[(1.0 / x), $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%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.1%

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

   1. associate-*r* 99.1%

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

   2. distribute-rgt-out 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 8: 99.7% 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 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 9: 2.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 31.8%

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

   1. associate-*r* 31.8%

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

   2. *-commutative 31.8%

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

   3. associate-*r/ 31.8%

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

   4. metadata-eval 31.8%

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

   5. rem-square-sqrt 31.8%

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

   6. fabs-sqr 31.8%

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

   7. rem-square-sqrt 31.8%

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

   8. pow-plus 31.8%

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

   9. metadata-eval 31.8%

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

7. Simplified 31.8%

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

8. Taylor expanded in x around 0 2.3%

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

   1. +-commutative 2.3%

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

   2. associate-*r* 2.3%

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

   3. associate-*r* 2.3%

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

   4. distribute-rgt-out 2.3%

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

   5. associate-*r/ 2.3%

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

   6. metadata-eval 2.3%

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

   7. associate-*r/ 2.3%

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

   8. metadata-eval 2.3%

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

10. Simplified 2.3%

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

11. Taylor expanded in x around inf 2.3%

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

    1. associate-*r* 2.3%

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

    2. associate-*r/ 2.3%

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

    3. metadata-eval 2.3%

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

13. Simplified 2.3%

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

    1. *-commutative 2.3%

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

    2. sqrt-div 2.3%

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

    3. metadata-eval 2.3%

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

    4. frac-times 2.3%

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

    5. metadata-eval 2.3%

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

15. Applied egg-rr 2.3%

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

16. Final simplification 2.3%

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

Alternative 10: 2.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

   3. distribute-rgt-out 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.1%

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

   1. associate-*r* 99.1%

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

   2. distribute-rgt-out 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in x around 0 2.4%

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

12. Taylor expanded in x around inf 2.3%

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

    1. associate-*l/ 2.3%

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

    2. *-lft-identity 2.3%

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

14. Simplified 2.3%

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

15. Final simplification 2.3%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}}}{x} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(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)))))))