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

Percentage Accurate: 100.0% → 100.0%

Time: 12.3s

Alternatives: 12

Speedup: 4.0×

• 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, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{{\left({\pi}^{0.25}\right)}^{2}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp (* x 2.0)) (* x 0.5)) (pow (pow PI 0.25) 2.0))
  (/
   (+
    1.0
    (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (pow x 2.0)) (/ 1.875 (pow x 6.0)))))
   x)))

C

double code(double x) {
	return (pow(exp((x * 2.0)), (x * 0.5)) / pow(pow(((double) M_PI), 0.25), 2.0)) * ((1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / pow(x, 2.0)) + (1.875 / pow(x, 6.0))))) / x);
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp((x * 2.0)), (x * 0.5)) / Math.pow(Math.pow(Math.PI, 0.25), 2.0)) * ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / Math.pow(x, 2.0)) + (1.875 / Math.pow(x, 6.0))))) / x);
}

Python

def code(x):
	return (math.pow(math.exp((x * 2.0)), (x * 0.5)) / math.pow(math.pow(math.pi, 0.25), 2.0)) * ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / math.pow(x, 2.0)) + (1.875 / math.pow(x, 6.0))))) / x)

Julia

function code(x)
	return Float64(Float64((exp(Float64(x * 2.0)) ^ Float64(x * 0.5)) / ((pi ^ 0.25) ^ 2.0)) * Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(1.875 / (x ^ 6.0))))) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((exp((x * 2.0)) ^ (x * 0.5)) / ((pi ^ 0.25) ^ 2.0)) * ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x ^ 2.0)) + (1.875 / (x ^ 6.0))))) / x);
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[Pi, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. sqr-pow 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. pow-prod-down 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. pow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
12. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x} \cdot e^{x}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. prod-exp 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x + x}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{x \cdot 1} + x}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 1 + \color{blue}{x \cdot 1}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    5. distribute-lft-out 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{x \cdot \left(1 + 1\right)}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    6. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot \color{blue}{2}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    7. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(\frac{\color{blue}{x \cdot 1}}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    8. associate-/l* 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    9. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot \color{blue}{0.5}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

13. Simplified 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
14. Step-by-step derivation

    1. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. pow2 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{\color{blue}{{\left(\sqrt{\sqrt{\pi}}\right)}^{2}}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. pow1/2 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{{\left(\sqrt{\color{blue}{{\pi}^{0.5}}}\right)}^{2}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. sqrt-pow1 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{{\color{blue}{\left({\pi}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{{\left({\pi}^{\color{blue}{0.25}}\right)}^{2}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

15. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{\color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
16. Add Preprocessing

Alternative 2: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    1.0
    (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (pow x 2.0)) (/ 1.875 (pow x 6.0)))))
   x)
  (/ (pow (exp (* x 2.0)) (* x 0.5)) (sqrt PI))))

C

double code(double x) {
	return ((1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / pow(x, 2.0)) + (1.875 / pow(x, 6.0))))) / x) * (pow(exp((x * 2.0)), (x * 0.5)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / Math.pow(x, 2.0)) + (1.875 / Math.pow(x, 6.0))))) / x) * (Math.pow(Math.exp((x * 2.0)), (x * 0.5)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / math.pow(x, 2.0)) + (1.875 / math.pow(x, 6.0))))) / x) * (math.pow(math.exp((x * 2.0)), (x * 0.5)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(1.875 / (x ^ 6.0))))) / x) * Float64((exp(Float64(x * 2.0)) ^ Float64(x * 0.5)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x ^ 2.0)) + (1.875 / (x ^ 6.0))))) / x) * ((exp((x * 2.0)) ^ (x * 0.5)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[Power[N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.5), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. sqr-pow 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. pow-prod-down 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. pow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
12. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x} \cdot e^{x}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. prod-exp 100.0%

       \[\leadsto \frac{{\color{blue}{\left(e^{x + x}\right)}}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{x \cdot 1} + x}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 1 + \color{blue}{x \cdot 1}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    5. distribute-lft-out 100.0%

       \[\leadsto \frac{{\left(e^{\color{blue}{x \cdot \left(1 + 1\right)}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    6. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot \color{blue}{2}}\right)}^{\left(\frac{x}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    7. *-rgt-identity 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(\frac{\color{blue}{x \cdot 1}}{2}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    8. associate-/l* 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    9. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot \color{blue}{0.5}\right)}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

13. Simplified 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

14. Final simplification 100.0%

    \[\leadsto \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{{\left(e^{x \cdot 2}\right)}^{\left(x \cdot 0.5\right)}}{\sqrt{\pi}} \]
15. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    1.0
    (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (pow x 2.0)) (/ 1.875 (pow x 6.0)))))
   x)
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return ((1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / pow(x, 2.0)) + (1.875 / pow(x, 6.0))))) / x) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / Math.pow(x, 2.0)) + (1.875 / Math.pow(x, 6.0))))) / x) * (Math.pow(Math.exp(x), x) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / math.pow(x, 2.0)) + (1.875 / math.pow(x, 6.0))))) / x) * (math.pow(math.exp(x), x) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(1.875 / (x ^ 6.0))))) / x) * Float64((exp(x) ^ x) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x ^ 2.0)) + (1.875 / (x ^ 6.0))))) / x) * ((exp(x) ^ x) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

12. Final simplification 100.0%

    \[\leadsto \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 4: 100.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    1.0
    (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (pow x 2.0)) (/ 1.875 (pow x 6.0)))))
   x)
  (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return ((1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / pow(x, 2.0)) + (1.875 / pow(x, 6.0))))) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / Math.pow(x, 2.0)) + (1.875 / Math.pow(x, 6.0))))) / x) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / math.pow(x, 2.0)) + (1.875 / math.pow(x, 6.0))))) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(1.875 / (x ^ 6.0))))) / x) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x ^ 2.0)) + (1.875 / (x ^ 6.0))))) / x) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]

10. Final simplification 99.9%

    \[\leadsto \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

9. Simplified 99.4%

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

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 6: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 7: 99.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

9. Simplified 99.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}} \]
10. Step-by-step derivation

    1. pow-exp 100.0%

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 99.3%

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

Alternative 8: 99.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \frac{0.5}{{x}^{2}}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

9. Simplified 99.3%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}} \]
10. Add Preprocessing

Alternative 9: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.2%

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

   1. pow-exp 100.0%

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

9. Applied egg-rr 99.2%

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

Alternative 10: 99.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[(x * 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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.2%

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

   1. un-div-inv 99.2%

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

   2. clear-num 99.2%

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

   3. associate-/l/ 99.2%

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

   4. pow2 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.2%

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

    2. associate-/r/ 99.2%

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

    3. associate-*l/ 99.2%

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

    4. *-lft-identity 99.2%

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{e^{{x}^{2}}}{x \cdot \sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 11: 99.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.2%

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

Alternative 12: 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. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)\right)}\right)}^{3}} \]

7. Taylor expanded in x around inf 99.2%

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

8. Taylor expanded in x around 0 2.3%

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

   1. associate-*l/ 2.3%

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

   2. *-lft-identity 2.3%

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

10. Simplified 2.3%

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

Reproduce

herbie shell --seed 2024156 
(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)))))))