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

Percentage Accurate: 100.0% → 100.0%

Time: 16.6s

Alternatives: 10

Speedup: 3.0×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))) (t_1 (pow t_0 3.0)) (t_2 (/ t_1 (fabs x))))
   (*
    (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0))
    (fma 1.875 (* t_1 t_2) (fma 0.75 (/ t_2 (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);
	double t_2 = t_1 / fabs(x);
	return (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0)) * fma(1.875, (t_1 * t_2), fma(0.75, (t_2 / 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
	t_2 = Float64(t_1 / abs(x))
	return Float64(Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)) * fma(1.875, Float64(t_1 * t_2), fma(0.75, Float64(t_2 / 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]}, Block[{t$95$2 = N[(t$95$1 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.875 * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(0.75 * N[(t$95$2 / 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. Step-by-step derivation

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := {t_0}^{3}\\ t_2 := \frac{t_1}{\left|x\right|}\\ \mathsf{fma}\left(1.875, t_1 \cdot t_2, \mathsf{fma}\left(0.75, \frac{t_2}{\left|x\right|}, \mathsf{fma}\left(0.5, t_1, t_0\right)\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))) (t_1 (pow t_0 3.0)) (t_2 (/ t_1 (fabs x))))
   (*
    (fma 1.875 (* t_1 t_2) (fma 0.75 (/ t_2 (fabs x)) (fma 0.5 t_1 t_0)))
    (/ (pow (exp x) x) (expm1 (log1p (sqrt PI)))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = pow(t_0, 3.0);
	double t_2 = t_1 / fabs(x);
	return fma(1.875, (t_1 * t_2), fma(0.75, (t_2 / fabs(x)), fma(0.5, t_1, t_0))) * (pow(exp(x), x) / expm1(log1p(sqrt(((double) M_PI)))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = t_0 ^ 3.0
	t_2 = Float64(t_1 / abs(x))
	return Float64(fma(1.875, Float64(t_1 * t_2), fma(0.75, Float64(t_2 / abs(x)), fma(0.5, t_1, t_0))) * Float64((exp(x) ^ x) / expm1(log1p(sqrt(pi)))))
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]}, Block[{t$95$2 = N[(t$95$1 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.875 * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(0.75 * N[(t$95$2 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(Exp[N[Log[1 + N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))) (t_1 (pow t_0 3.0)) (t_2 (/ t_1 (fabs x))))
   (*
    (fma 1.875 (* t_1 t_2) (fma 0.75 (/ t_2 (fabs x)) (fma 0.5 t_1 t_0)))
    (/ (pow (exp x) x) (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = pow(t_0, 3.0);
	double t_2 = t_1 / fabs(x);
	return fma(1.875, (t_1 * t_2), fma(0.75, (t_2 / fabs(x)), fma(0.5, t_1, t_0))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = t_0 ^ 3.0
	t_2 = Float64(t_1 / abs(x))
	return Float64(fma(1.875, Float64(t_1 * t_2), fma(0.75, Float64(t_2 / abs(x)), fma(0.5, t_1, t_0))) * Float64((exp(x) ^ x) / sqrt(pi)))
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]}, Block[{t$95$2 = N[(t$95$1 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.875 * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(0.75 * N[(t$95$2 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) * (sqrt((1.0 / pi)) * (((1.875 / (x ^ 7.0)) + (1.0 / x)) + ((0.75 / (x ^ 5.0)) + (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[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. associate-+r+ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-*r/ 100.0%

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

   7. metadata-eval 100.0%

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

8. Simplified 100.0%

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

   1. exp-prod 100.0%

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

10. Applied egg-rr 100.0%

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

11. Taylor expanded in x around 0 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{1}{{x}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{1}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
12. 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. +-commutative 100.0%

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

    3. associate-+l+ 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (math.sqrt((1.0 / math.pi)) * (((1.875 / math.pow(x, 7.0)) + (1.0 / x)) + ((0.75 / math.pow(x, 5.0)) + (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(Float64(1.875 / (x ^ 7.0)) + Float64(1.0 / x)) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(0.5 / (x ^ 3.0))))) * exp(Float64(x * x)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. associate-+r+ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-*r/ 100.0%

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

   7. metadata-eval 100.0%

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

8. Simplified 100.0%

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

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. +-commutative 100.0%

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

    3. associate-+l+ 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 6: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

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) + (0.5 / Math.pow(x, 3.0)))));
}

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) + (0.5 / math.pow(x, 3.0)))))

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(0.5 / (x ^ 3.0))))))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * (sqrt((1.0 / pi)) * ((0.75 / (x ^ 5.0)) + ((1.0 / x) + (0.5 / (x ^ 3.0)))));
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[(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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. associate-+r+ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-*r/ 100.0%

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

   7. metadata-eval 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around inf 99.8%

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

    1. +-commutative 99.8%

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

    2. associate-+l+ 99.8%

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

    3. associate-*r* 99.8%

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

    4. *-commutative 99.8%

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

    5. associate-*r* 99.8%

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

    6. distribute-rgt-out 99.8%

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

    7. distribute-lft-out 99.8%

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

    8. associate-*r/ 99.8%

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

    9. metadata-eval 99.8%

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

    10. associate-*r/ 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 7: 99.6% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Power[Pi, -0.5], $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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.8%

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

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

8. Simplified 99.8%

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

   1. distribute-lft-in 99.8%

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

   2. div-inv 99.8%

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

   3. inv-pow 99.8%

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

   4. sqrt-pow1 99.8%

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

   5. metadata-eval 99.8%

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

   6. inv-pow 99.8%

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

   7. sqrt-pow1 99.8%

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

   8. metadata-eval 99.8%

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

   9. div-inv 99.8%

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

   10. pow-flip 99.8%

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

   11. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

    1. *-lft-identity 99.8%

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

    2. associate-*r/ 99.8%

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

    3. associate-*l/ 99.8%

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

    4. *-commutative 99.8%

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

    5. distribute-lft-in 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 8: 99.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.7%

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

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

8. Simplified 99.7%

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

   1. expm1-log1p-u 2.3%

      \[\leadsto 1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\frac{1}{\pi}}}{x}\right)\right)} \]

   2. expm1-udef 1.7%

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

   3. inv-pow 1.7%

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

   4. sqrt-pow1 1.7%

      \[\leadsto 1 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right)} - 1\right) \]

   5. metadata-eval 1.7%

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

10. Applied egg-rr 4.8%

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

    1. expm1-def 2.3%

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

    2. expm1-log1p 2.3%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \]

Alternative 9: 5.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x + N[(1.0 / 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. Step-by-step derivation

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.7%

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

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 5.5%

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

    1. distribute-rgt-out 5.5%

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

11. Simplified 5.5%

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

12. Final simplification 5.5%

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

Alternative 10: 2.3% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.7%

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

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 2.3%

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

    1. expm1-log1p-u 2.3%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\frac{1}{\pi}}}{x}\right)\right)} \]

    2. expm1-udef 1.7%

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

    3. inv-pow 1.7%

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

    4. sqrt-pow1 1.7%

       \[\leadsto 1 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right)} - 1\right) \]

    5. metadata-eval 1.7%

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

11. Applied egg-rr 1.7%

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

    1. expm1-def 2.3%

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

    2. expm1-log1p 2.3%

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

13. Simplified 2.3%

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

14. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2023322 
(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)))))))