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

Percentage Accurate: 100.0% → 100.0%

Time: 29.0s

Alternatives: 8

Speedup: 2.9×

• 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := {t\_0}^{3}\\ t_2 := \frac{t\_1}{\left|x\right|}\\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(1.875, t\_1 \cdot t\_2, \mathsf{fma}\left(0.75, \frac{t\_2}{\left|x\right|}, \mathsf{fma}\left(0.5, t\_1, t\_0\right)\right)\right) \end{array} \end{array} \]

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) (cbrt (pow PI 1.5)))
    (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) / cbrt(pow(((double) M_PI), 1.5))) * 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((pi ^ 1.5))) * 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[Pi, 1.5], $MachinePrecision], 1/3], $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 99.9%

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

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. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. pow1 100.0%

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

   5. pow1/2 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. unpow1/3 100.0%

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

8. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{\frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{\left|x\right|}}{\left|x\right|}, \mathsf{fma}\left(0.5, {\left(\frac{1}{\left|x\right|}\right)}^{3}, \frac{1}{\left|x\right|}\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (cbrt (pow PI 1.5)))
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	return (pow(exp(x), x) / cbrt(pow(((double) M_PI), 1.5))) * fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / cbrt((pi ^ 1.5))) * fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

   1. exp-prod 100.0%

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

10. Applied egg-rr 100.0%

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

    1. add-cbrt-cube 100.0%

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

    2. pow1/3 100.0%

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

    3. add-sqr-sqrt 100.0%

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

    4. pow1 100.0%

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

    5. pow1/2 100.0%

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

    6. pow-prod-up 100.0%

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

    7. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
13. Step-by-step derivation

    1. unpow1/3 100.0%

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

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
15. Step-by-step derivation

    1. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. *-un-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot \left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    8. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    9. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    10. inv-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 2\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    11. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left({\left(\left|x\right|\right)}^{-1}\right)}^{\color{blue}{5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    12. pow-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    13. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    14. fabs-sqr 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    15. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{x}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    16. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{\color{blue}{-5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

16. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5} \cdot 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

17. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
18. Add Preprocessing

Alternative 3: 100.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

   1. exp-prod 100.0%

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

10. Applied egg-rr 100.0%

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

    1. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. *-un-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot \left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    8. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    9. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    10. inv-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 2\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    11. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left({\left(\left|x\right|\right)}^{-1}\right)}^{\color{blue}{5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    12. pow-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    13. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    14. fabs-sqr 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    15. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{x}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    16. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{\color{blue}{-5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

   1. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. pow-prod-up 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. *-un-lft-identity 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot \left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. *-commutative 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. pow3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   8. pow2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   9. pow-prod-up 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   10. inv-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 2\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left({\left(\left|x\right|\right)}^{-1}\right)}^{\color{blue}{5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   12. pow-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   13. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   14. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   15. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{x}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{\color{blue}{-5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 99.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

   1. exp-prod 100.0%

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

10. Applied egg-rr 100.0%

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

    1. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    2. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    3. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{2}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    4. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    5. *-un-lft-identity 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot \left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    6. *-commutative 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    7. pow3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    8. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    9. pow-prod-up 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    10. inv-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 2\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    11. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left({\left(\left|x\right|\right)}^{-1}\right)}^{\color{blue}{5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    12. pow-pow 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    13. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    14. fabs-sqr 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    15. add-sqr-sqrt 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {\color{blue}{x}}^{\left(-1 \cdot 5\right)} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

    16. metadata-eval 100.0%

        \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(0.75, {x}^{\color{blue}{-5}} \cdot 1, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

12. Applied egg-rr 100.0%

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

13. Taylor expanded in x around inf 98.8%

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

    1. associate-*r/ 98.8%

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

    2. *-commutative 98.8%

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

    3. associate-/l* 98.8%

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

15. Simplified 98.8%

    \[\leadsto \color{blue}{e^{{x}^{2}} \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}} \]
16. Add Preprocessing

Alternative 6: 2.3% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 2.4%

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

    1. expm1-log1p-u 2.4%

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

    2. expm1-undefine 1.8%

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

11. Applied egg-rr 1.8%

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

    1. sub-neg 1.8%

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

    2. metadata-eval 1.8%

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

    3. +-commutative 1.8%

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

    4. log1p-undefine 1.8%

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

    5. rem-exp-log 1.8%

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

    6. associate-+r+ 2.4%

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

    7. metadata-eval 2.4%

       \[\leadsto \color{blue}{0} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    8. metadata-eval 2.4%

       \[\leadsto \color{blue}{\left(0 - 0\right)} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    9. associate--r- 2.4%

       \[\leadsto \color{blue}{0 - \left(0 - \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    10. neg-sub0 2.4%

        \[\leadsto 0 - \color{blue}{\left(-\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    11. distribute-frac-neg 2.4%

        \[\leadsto 0 - \color{blue}{\frac{-\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

13. Simplified 2.4%

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

14. Taylor expanded in x around inf 2.4%

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

16. Simplified 2.4%

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

Alternative 7: 2.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 2.4%

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

    1. expm1-log1p-u 2.4%

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

    2. expm1-undefine 1.8%

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

11. Applied egg-rr 1.8%

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

    1. sub-neg 1.8%

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

    2. metadata-eval 1.8%

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

    3. +-commutative 1.8%

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

    4. log1p-undefine 1.8%

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

    5. rem-exp-log 1.8%

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

    6. associate-+r+ 2.4%

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

    7. metadata-eval 2.4%

       \[\leadsto \color{blue}{0} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    8. metadata-eval 2.4%

       \[\leadsto \color{blue}{\left(0 - 0\right)} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    9. associate--r- 2.4%

       \[\leadsto \color{blue}{0 - \left(0 - \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    10. neg-sub0 2.4%

        \[\leadsto 0 - \color{blue}{\left(-\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    11. distribute-frac-neg 2.4%

        \[\leadsto 0 - \color{blue}{\frac{-\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

13. Simplified 2.4%

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

14. Taylor expanded in x around inf 2.4%

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

    1. associate-*r* 2.4%

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

    2. distribute-rgt1-in 2.4%

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

    3. +-commutative 2.4%

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

    4. associate-*r/ 2.4%

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

    5. metadata-eval 2.4%

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

16. Simplified 2.4%

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

17. Final simplification 2.4%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 + \frac{0.5}{{x}^{2}}\right)}{x} \]
18. Add Preprocessing

Alternative 8: 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 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 2.4%

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

    1. expm1-log1p-u 2.4%

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

    2. expm1-undefine 1.8%

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

11. Applied egg-rr 1.8%

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

    1. sub-neg 1.8%

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

    2. metadata-eval 1.8%

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

    3. +-commutative 1.8%

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

    4. log1p-undefine 1.8%

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

    5. rem-exp-log 1.8%

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

    6. associate-+r+ 2.4%

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

    7. metadata-eval 2.4%

       \[\leadsto \color{blue}{0} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    8. metadata-eval 2.4%

       \[\leadsto \color{blue}{\left(0 - 0\right)} + \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}} \]

    9. associate--r- 2.4%

       \[\leadsto \color{blue}{0 - \left(0 - \frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    10. neg-sub0 2.4%

        \[\leadsto 0 - \color{blue}{\left(-\frac{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}\right)} \]

    11. distribute-frac-neg 2.4%

        \[\leadsto 0 - \color{blue}{\frac{-\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}{\sqrt{\pi}}} \]

13. Simplified 2.4%

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

14. Taylor expanded in x around inf 2.4%

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

    1. associate-*l/ 2.4%

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

    2. *-lft-identity 2.4%

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

16. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{x}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024093 
(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)))))))