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

Percentage Accurate: 100.0% → 100.0%

Time: 18.5s

Alternatives: 8

Speedup: 3.3×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. metadata-eval 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. metadata-eval 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-*r/ 99.5%

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

   5. metadata-eval 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. metadata-eval 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-*r/ 99.5%

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

   5. metadata-eval 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around 0 99.5%

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

    1. +-commutative 99.5%

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

    2. associate-+l+ 99.5%

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

    3. +-commutative 99.5%

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

    4. associate-*r/ 99.5%

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

    5. metadata-eval 99.5%

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

    6. associate-*r/ 99.5%

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

    7. metadata-eval 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. metadata-eval 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-*r/ 99.5%

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

   5. metadata-eval 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

9. Simplified 99.4%

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

    1. rem-cube-cbrt 99.4%

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

    2. inv-pow 99.4%

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

    3. sqrt-pow1 99.4%

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

    4. metadata-eval 99.4%

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

    5. *-un-lft-identity 99.4%

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

    6. *-un-lft-identity 99.4%

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

    7. times-frac 99.4%

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

    8. metadata-eval 99.4%

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

11. Applied egg-rr 99.4%

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

    1. *-lft-identity 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 6: 2.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. metadata-eval 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-*r/ 99.5%

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

   5. metadata-eval 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in x around 0 2.3%

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

14. Final simplification 2.3%

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

Alternative 7: 1.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.875 (/ (pow PI -0.5) (pow x 7.0))))

C

double code(double x) {
	return 1.875 * (pow(((double) M_PI), -0.5) / pow(x, 7.0));
}

Java

public static double code(double x) {
	return 1.875 * (Math.pow(Math.PI, -0.5) / Math.pow(x, 7.0));
}

Python

def code(x):
	return 1.875 * (math.pow(math.pi, -0.5) / math.pow(x, 7.0))

Julia

function code(x)
	return Float64(1.875 * Float64((pi ^ -0.5) / (x ^ 7.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.875 * ((pi ^ -0.5) / (x ^ 7.0));
end

Wolfram

code[x_] := N[(1.875 * N[(N[Power[Pi, -0.5], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 13.0%

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

8. Taylor expanded in x around 0 0.9%

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

   1. +-commutative 0.9%

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

   2. unpow2 0.9%

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

   3. fma-define 0.9%

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

10. Simplified 0.9%

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

11. Taylor expanded in x around 0 1.7%

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

    1. associate-*l/ 1.7%

       \[\leadsto 1.875 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{7}}} \]

    2. *-lft-identity 1.7%

       \[\leadsto 1.875 \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{{x}^{7}} \]

    3. unpow-1 1.7%

       \[\leadsto 1.875 \cdot \frac{\sqrt{\color{blue}{{\pi}^{-1}}}}{{x}^{7}} \]

    4. metadata-eval 1.7%

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

    5. pow-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}}{{x}^{7}} \]

    6. rem-sqrt-square 1.7%

       \[\leadsto 1.875 \cdot \frac{\color{blue}{\left|{\pi}^{-0.5}\right|}}{{x}^{7}} \]

    7. metadata-eval 1.7%

       \[\leadsto 1.875 \cdot \frac{\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|}{{x}^{7}} \]

    8. pow-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right|}{{x}^{7}} \]

    9. fabs-sqr 1.7%

       \[\leadsto 1.875 \cdot \frac{\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}}{{x}^{7}} \]

    10. pow-sqr 1.7%

        \[\leadsto 1.875 \cdot \frac{\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}}{{x}^{7}} \]

    11. metadata-eval 1.7%

        \[\leadsto 1.875 \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{{x}^{7}} \]

13. Simplified 1.7%

    \[\leadsto \color{blue}{1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}}} \]

14. Final simplification 1.7%

    \[\leadsto 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{7}} \]
15. Add Preprocessing

Alternative 8: 1.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.875 (/ (pow PI -0.5) (pow x 5.0))))

C

double code(double x) {
	return 1.875 * (pow(((double) M_PI), -0.5) / pow(x, 5.0));
}

Java

public static double code(double x) {
	return 1.875 * (Math.pow(Math.PI, -0.5) / Math.pow(x, 5.0));
}

Python

def code(x):
	return 1.875 * (math.pow(math.pi, -0.5) / math.pow(x, 5.0))

Julia

function code(x)
	return Float64(1.875 * Float64((pi ^ -0.5) / (x ^ 5.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.875 * ((pi ^ -0.5) / (x ^ 5.0));
end

Wolfram

code[x_] := N[(1.875 * N[(N[Power[Pi, -0.5], $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 13.0%

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

8. Taylor expanded in x around 0 0.9%

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

   1. +-commutative 0.9%

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

   2. unpow2 0.9%

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

   3. fma-define 0.9%

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

10. Simplified 0.9%

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

11. Taylor expanded in x around inf 1.8%

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

    1. *-commutative 1.8%

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

    2. associate-*l/ 1.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{\pi}}}{{x}^{5}}} \cdot 1.875 \]

    3. *-lft-identity 1.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\pi}}}}{{x}^{5}} \cdot 1.875 \]

    4. unpow-1 1.8%

       \[\leadsto \frac{\sqrt{\color{blue}{{\pi}^{-1}}}}{{x}^{5}} \cdot 1.875 \]

    5. metadata-eval 1.8%

       \[\leadsto \frac{\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{{x}^{5}} \cdot 1.875 \]

    6. pow-sqr 1.8%

       \[\leadsto \frac{\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}}{{x}^{5}} \cdot 1.875 \]

    7. rem-sqrt-square 1.8%

       \[\leadsto \frac{\color{blue}{\left|{\pi}^{-0.5}\right|}}{{x}^{5}} \cdot 1.875 \]

    8. metadata-eval 1.8%

       \[\leadsto \frac{\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|}{{x}^{5}} \cdot 1.875 \]

    9. pow-sqr 1.8%

       \[\leadsto \frac{\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right|}{{x}^{5}} \cdot 1.875 \]

    10. fabs-sqr 1.8%

        \[\leadsto \frac{\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}}{{x}^{5}} \cdot 1.875 \]

    11. pow-sqr 1.8%

        \[\leadsto \frac{\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}}{{x}^{5}} \cdot 1.875 \]

    12. metadata-eval 1.8%

        \[\leadsto \frac{{\pi}^{\color{blue}{-0.5}}}{{x}^{5}} \cdot 1.875 \]

13. Simplified 1.8%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{{x}^{5}} \cdot 1.875} \]

14. Final simplification 1.8%

    \[\leadsto 1.875 \cdot \frac{{\pi}^{-0.5}}{{x}^{5}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))))))