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

Percentage Accurate: 100.0% → 99.9%

Time: 18.6s

Alternatives: 20

Speedup: 9.2×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (exp -1.0)
  (log
   (/
    (/ (* (sqrt PI) x) (exp (* x x)))
    (+ 1.0 (/ (+ 0.5 (/ (+ 0.75 (/ 1.875 (* x x))) (* x x))) (* x x)))))))

C

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

Java

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

Python

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

Julia

function code(x)
	return exp(-1.0) ^ log(Float64(Float64(Float64(sqrt(pi) * x) / exp(Float64(x * x))) / Float64(1.0 + Float64(Float64(0.5 + Float64(Float64(0.75 + Float64(1.875 / Float64(x * x))) / Float64(x * x))) / Float64(x * x)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right)}^{\color{blue}{-1}} \]

   3. pow-to-exp N/A

      \[\leadsto e^{\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right) \cdot -1} \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right) \cdot -1\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right), -1\right)\right) \]

8. Applied egg-rr 100.0%

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

   1. *-commutative N/A

      \[\leadsto e^{-1 \cdot \log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)} \]

   2. exp-prod N/A

      \[\leadsto {\left(e^{-1}\right)}^{\color{blue}{\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}} \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(e^{-1}\right), \color{blue}{\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \color{blue}{\left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

   5. rem-exp-log N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \left(e^{\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}\right)\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\left(e^{\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}\right)\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left(e^{-1}\right)}^{\log \left(\frac{\frac{\sqrt{\pi} \cdot x}{e^{x \cdot x}}}{1 + \frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}} \]
11. Add Preprocessing

Alternative 2: 99.9% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (exp
  (log
   (/
    (+ 1.0 (/ (+ 0.5 (/ (+ 0.75 (/ 1.875 (* x x))) (* x x))) (* x x)))
    (/ (* (sqrt PI) x) (exp (* x x)))))))

C

double code(double x) {
	return exp(log(((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / ((sqrt(((double) M_PI)) * x) / exp((x * x))))));
}

Java

public static double code(double x) {
	return Math.exp(Math.log(((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / ((Math.sqrt(Math.PI) * x) / Math.exp((x * x))))));
}

Python

def code(x):
	return math.exp(math.log(((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / ((math.sqrt(math.pi) * x) / math.exp((x * x))))))

Julia

function code(x)
	return exp(log(Float64(Float64(1.0 + Float64(Float64(0.5 + Float64(Float64(0.75 + Float64(1.875 / Float64(x * x))) / Float64(x * x))) / Float64(x * x))) / Float64(Float64(sqrt(pi) * x) / exp(Float64(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = exp(log(((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / ((sqrt(pi) * x) / exp((x * x))))));
end

Wolfram

code[x_] := N[Exp[N[Log[N[(N[(1.0 + N[(N[(0.5 + N[(N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[Pi], $MachinePrecision] * x), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right)}^{\color{blue}{-1}} \]

   3. pow-to-exp N/A

      \[\leadsto e^{\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right) \cdot -1} \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right) \cdot -1\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)}\right), -1\right)\right) \]

8. Applied egg-rr 100.0%

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

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right) \cdot -1\right)\right) \]

   2. rem-log-exp N/A

      \[\leadsto \mathsf{exp.f64}\left(\log \left(e^{\log \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right) \cdot -1}\right)\right) \]

   3. exp-to-pow N/A

      \[\leadsto \mathsf{exp.f64}\left(\log \left({\left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}\right)}^{-1}\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \mathsf{exp.f64}\left(\log \left(\frac{1}{\frac{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{exp.f64}\left(\log \left(\frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}\right)\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{e^{x \cdot x}}{x}}}\right)\right)\right) \]

10. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left({\mathsf{PI}\left(\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. add-sqr-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left({\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left({\left(\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left({\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{pow.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{4}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 9.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((exp((x * x)) * (1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x)))) / x) / sqrt(((double) M_PI));
}

Java

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

Python

def code(x):
	return ((math.exp((x * x)) * (1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x)))) / x) / math.sqrt(math.pi)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

8. Applied egg-rr 100.0%

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

Alternative 5: 99.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\sqrt{\pi} \cdot \frac{x}{e^{x \cdot x}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{3}{4}}{{x}^{2}}\right)}\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left({x}^{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\frac{x}{e^{x \cdot x}}}}{\sqrt{\pi}}} \]
12. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{x}{e^{x \cdot x}}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{\color{blue}{e^{x \cdot x}}}} \]

    3. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right)\right)}{\color{blue}{\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)}\right) \]

    5. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(0 - \left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}}\right)\right)\right) \]

    6. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}}\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{\color{blue}{e^{x \cdot x}}}\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}\right), \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{3}{4}}{x \cdot x}\right)\right), \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot x}{e^{x \cdot x}}\right)\right)\right) \]

13. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{0 - \left(1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\sqrt{\pi} \cdot \frac{0 - x}{e^{x \cdot x}}}} \]

14. Final simplification 99.7%

    \[\leadsto \frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\sqrt{\pi} \cdot \frac{x}{e^{x \cdot x}}} \]
15. Add Preprocessing

Alternative 6: 99.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\frac{x}{e^{x \cdot x}}}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{3}{4}}{{x}^{2}}\right)}\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left({x}^{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\frac{x}{e^{x \cdot x}}}}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 7: 99.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{3}{4}}{{x}^{2}}\right)}\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left({x}^{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4}}{x \cdot x}}{x \cdot x}\right) \cdot \left(e^{x \cdot x} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\frac{1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}{\frac{x}{e^{x \cdot x}}}}{\sqrt{\pi}}} \]

12. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
13. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

14. Simplified 99.7%

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

Alternative 8: 99.6% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

Alternative 9: 91.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.5\right)\\ t_1 := 1 + t\_0\\ t_2 := \left(x \cdot x\right) \cdot t\_1\\ t_3 := \left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_2\right)\\ t_4 := t\_2 + -1\\ \mathbf{if}\;x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\frac{1 + t\_2 \cdot t\_3}{1 + t\_2 \cdot t\_4}}{x}}{\sqrt{\pi}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{t\_3 + -1}{t\_4}}{x}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot t\_0\right)}{x}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.5)))
        (t_1 (+ 1.0 t_0))
        (t_2 (* (* x x) t_1))
        (t_3 (* (* x x) (* t_1 t_2)))
        (t_4 (+ t_2 -1.0)))
   (if (<= x 4e+38)
     (/ (/ (/ (+ 1.0 (* t_2 t_3)) (+ 1.0 (* t_2 t_4))) x) (sqrt PI))
     (if (<= x 1.35e+77)
       (/ (/ (/ (+ t_3 -1.0) t_4) x) (sqrt PI))
       (/ (/ (* x (* x t_0)) x) (sqrt PI))))))

C

double code(double x) {
	double t_0 = x * (x * 0.5);
	double t_1 = 1.0 + t_0;
	double t_2 = (x * x) * t_1;
	double t_3 = (x * x) * (t_1 * t_2);
	double t_4 = t_2 + -1.0;
	double tmp;
	if (x <= 4e+38) {
		tmp = (((1.0 + (t_2 * t_3)) / (1.0 + (t_2 * t_4))) / x) / sqrt(((double) M_PI));
	} else if (x <= 1.35e+77) {
		tmp = (((t_3 + -1.0) / t_4) / x) / sqrt(((double) M_PI));
	} else {
		tmp = ((x * (x * t_0)) / x) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * 0.5);
	double t_1 = 1.0 + t_0;
	double t_2 = (x * x) * t_1;
	double t_3 = (x * x) * (t_1 * t_2);
	double t_4 = t_2 + -1.0;
	double tmp;
	if (x <= 4e+38) {
		tmp = (((1.0 + (t_2 * t_3)) / (1.0 + (t_2 * t_4))) / x) / Math.sqrt(Math.PI);
	} else if (x <= 1.35e+77) {
		tmp = (((t_3 + -1.0) / t_4) / x) / Math.sqrt(Math.PI);
	} else {
		tmp = ((x * (x * t_0)) / x) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.5)
	t_1 = 1.0 + t_0
	t_2 = (x * x) * t_1
	t_3 = (x * x) * (t_1 * t_2)
	t_4 = t_2 + -1.0
	tmp = 0
	if x <= 4e+38:
		tmp = (((1.0 + (t_2 * t_3)) / (1.0 + (t_2 * t_4))) / x) / math.sqrt(math.pi)
	elif x <= 1.35e+77:
		tmp = (((t_3 + -1.0) / t_4) / x) / math.sqrt(math.pi)
	else:
		tmp = ((x * (x * t_0)) / x) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.5))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(x * x) * t_1)
	t_3 = Float64(Float64(x * x) * Float64(t_1 * t_2))
	t_4 = Float64(t_2 + -1.0)
	tmp = 0.0
	if (x <= 4e+38)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(t_2 * t_3)) / Float64(1.0 + Float64(t_2 * t_4))) / x) / sqrt(pi));
	elseif (x <= 1.35e+77)
		tmp = Float64(Float64(Float64(Float64(t_3 + -1.0) / t_4) / x) / sqrt(pi));
	else
		tmp = Float64(Float64(Float64(x * Float64(x * t_0)) / x) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.5);
	t_1 = 1.0 + t_0;
	t_2 = (x * x) * t_1;
	t_3 = (x * x) * (t_1 * t_2);
	t_4 = t_2 + -1.0;
	tmp = 0.0;
	if (x <= 4e+38)
		tmp = (((1.0 + (t_2 * t_3)) / (1.0 + (t_2 * t_4))) / x) / sqrt(pi);
	elseif (x <= 1.35e+77)
		tmp = (((t_3 + -1.0) / t_4) / x) / sqrt(pi);
	else
		tmp = ((x * (x * t_0)) / x) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + -1.0), $MachinePrecision]}, If[LessEqual[x, 4e+38], N[(N[(N[(N[(1.0 + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+77], N[(N[(N[(N[(t$95$3 + -1.0), $MachinePrecision] / t$95$4), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.99999999999999991e38

   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%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      2. associate-/r* N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      4. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 97.6%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 4.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 4.0%

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

       1. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{1}^{3} + {\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({1}^{3} + {\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)}^{3}\right), \left(1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   14. Applied egg-rr 43.0%

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

   if 3.99999999999999991e38 < x < 1.3499999999999999e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      2. associate-/r* N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 5.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 5.0%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) + 1\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) - 1}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot 1\right), \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) - 1\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   14. Applied egg-rr 100.0%

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

   if 1.3499999999999999e77 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      2. associate-/r* N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {x}^{4}\right)}, x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   14. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       19. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   15. Simplified 100.0%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)}}{x}}{\sqrt{\pi}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right)\right)\right)}{1 + \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right) + -1\right)}}{x}}{\sqrt{\pi}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{\left(x \cdot x\right) \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right)\right) + -1}{\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right) + -1}}{x}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)}{x}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.5\right)\\ t_1 := 1 + t\_0\\ t_2 := \left(x \cdot x\right) \cdot t\_1\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{\left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_2\right) + -1}{t\_2 + -1}}{x}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot t\_0\right)}{x}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.5))) (t_1 (+ 1.0 t_0)) (t_2 (* (* x x) t_1)))
   (if (<= x 1.35e+77)
     (/ (/ (/ (+ (* (* x x) (* t_1 t_2)) -1.0) (+ t_2 -1.0)) x) (sqrt PI))
     (/ (/ (* x (* x t_0)) x) (sqrt PI)))))

C

double code(double x) {
	double t_0 = x * (x * 0.5);
	double t_1 = 1.0 + t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1.35e+77) {
		tmp = (((((x * x) * (t_1 * t_2)) + -1.0) / (t_2 + -1.0)) / x) / sqrt(((double) M_PI));
	} else {
		tmp = ((x * (x * t_0)) / x) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * 0.5);
	double t_1 = 1.0 + t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1.35e+77) {
		tmp = (((((x * x) * (t_1 * t_2)) + -1.0) / (t_2 + -1.0)) / x) / Math.sqrt(Math.PI);
	} else {
		tmp = ((x * (x * t_0)) / x) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.5)
	t_1 = 1.0 + t_0
	t_2 = (x * x) * t_1
	tmp = 0
	if x <= 1.35e+77:
		tmp = (((((x * x) * (t_1 * t_2)) + -1.0) / (t_2 + -1.0)) / x) / math.sqrt(math.pi)
	else:
		tmp = ((x * (x * t_0)) / x) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.5))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if (x <= 1.35e+77)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(x * x) * Float64(t_1 * t_2)) + -1.0) / Float64(t_2 + -1.0)) / x) / sqrt(pi));
	else
		tmp = Float64(Float64(Float64(x * Float64(x * t_0)) / x) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.5);
	t_1 = 1.0 + t_0;
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if (x <= 1.35e+77)
		tmp = (((((x * x) * (t_1 * t_2)) + -1.0) / (t_2 + -1.0)) / x) / sqrt(pi);
	else
		tmp = ((x * (x * t_0)) / x) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, 1.35e+77], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3499999999999999e77

   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 99.9%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      2. associate-/r* N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      4. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 98.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 4.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 4.5%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) + 1\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) - 1}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) - 1 \cdot 1\right), \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) - 1\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   14. Applied egg-rr 51.2%

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

   if 1.3499999999999999e77 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      2. associate-/r* N/A

         \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {x}^{4}\right)}, x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
   14. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       19. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   15. Simplified 100.0%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)}}{x}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{\left(x \cdot x\right) \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right)\right) + -1}{\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right) + -1}}{x}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)}{x}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.1% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{x} \cdot \frac{1 + \frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    1.0
    (* x (* x (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.16666666666666666))))))))
   x)
  (/
   (+ 1.0 (/ (+ 0.5 (/ (+ 0.75 (/ 1.875 (* x x))) (* x x))) (* x x)))
   (sqrt PI))))

C

double code(double x) {
	return ((1.0 + (x * (x * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666)))))))) / x) * ((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return ((1.0 + (x * (x * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666)))))))) / x) * ((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((1.0 + (x * (x * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666)))))))) / x) * ((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666)))))))) / x) * Float64(Float64(1.0 + Float64(Float64(0.5 + Float64(Float64(0.75 + Float64(1.875 / Float64(x * x))) / Float64(x * x))) / Float64(x * x))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + (x * (x * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666)))))))) / x) * ((1.0 + ((0.5 + ((0.75 + (1.875 / (x * x))) / (x * x))) / (x * x))) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.5 + N[(N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 82.9%

   \[\leadsto \frac{\left(1 + \frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}{x} \cdot \left(1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}{x} \cdot \color{blue}{\frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}{x}\right), \color{blue}{\left(\frac{1 + \frac{\frac{1}{2} + \frac{\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

11. Applied egg-rr 82.9%

    \[\leadsto \color{blue}{\frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{x} \cdot \frac{1 + \frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x}}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 12: 84.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right) \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (+ 1.0 (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)))
   (/
    (+
     1.0
     (* (* x x) (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666)))))))
    x))
  (sqrt PI)))

C

double code(double x) {
	return ((1.0 + ((0.5 + (0.75 / (x * x))) / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((1.0 + ((0.5 + (0.75 / (x * x))) / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((1.0 + ((0.5 + (0.75 / (x * x))) / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x))) * Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))))) / x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((0.5 + (0.75 / (x * x))) / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{3}{4}}{{x}^{2}}\right)}\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left({x}^{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 82.9%

    \[\leadsto \frac{\left(1 + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right) \cdot \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 13: 84.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{0.5}{x \cdot x}\right) \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (+ 1.0 (/ 0.5 (* x x)))
   (/
    (+
     1.0
     (* (* x x) (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666)))))))
    x))
  (sqrt PI)))

C

double code(double x) {
	return ((1.0 + (0.5 / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((1.0 + (0.5 / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((1.0 + (0.5 / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) * Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))))) / x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + (0.5 / (x * x))) * ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 82.9%

   \[\leadsto \frac{\left(1 + \frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x}\right) \cdot \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}}{\sqrt{\pi}} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. *-lowering-*.f64 82.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 82.9%

    \[\leadsto \frac{\color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right)} \cdot \frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 14: 84.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (+
    1.0
    (* (* x x) (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666)))))))
   x)
  (sqrt PI)))

C

double code(double x) {
	return ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))))) / x) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((x * x) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))) / x) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

12. Simplified 82.9%

    \[\leadsto \frac{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)}{x}}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 15: 76.3% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. *-lowering-*.f64 76.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 76.5%

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

    1. associate-/l/ N/A

       \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot x}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot x\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{x}\right)\right) \]

    12. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), x\right)\right) \]

    13. PI-lowering-PI.f64 76.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), x\right)\right) \]

14. Applied egg-rr 76.5%

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

Alternative 16: 76.4% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. *-lowering-*.f64 76.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 76.5%

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

13. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {x}^{4}\right)}, x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
14. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. unpow3 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{3}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    12. unpow3 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    14. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    17. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    18. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    19. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    20. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    21. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    22. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    23. *-lowering-*.f64 76.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

15. Simplified 76.5%

    \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)}}{x}}{\sqrt{\pi}} \]
16. Add Preprocessing

Alternative 17: 68.5% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) + x \cdot 1.25}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* x (* (* x x) 0.5)) (* x 1.25)) (sqrt PI)))

C

double code(double x) {
	return ((x * ((x * x) * 0.5)) + (x * 1.25)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return ((x * ((x * x) * 0.5)) + (x * 1.25)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return ((x * ((x * x) * 0.5)) + (x * 1.25)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(x * Float64(Float64(x * x) * 0.5)) + Float64(x * 1.25)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = ((x * ((x * x) * 0.5)) + (x * 1.25)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 1.25), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot x\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   16. *-lowering-*.f64 76.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 76.5%

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

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{2} + \frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} \cdot \frac{1}{2} + {x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{3} + {x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {x}^{3}\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    4. unpow3 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left({x}^{3} \cdot \left(\frac{5}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left({x}^{3} \cdot \left(\frac{1}{{x}^{2}} \cdot \frac{5}{4}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    14. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(\left({x}^{3} \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    15. cube-mult N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(\left(\left(x \cdot {x}^{2}\right) \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    17. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(\left(x \cdot \left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    18. rgt-mult-inverse N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(\left(x \cdot 1\right) \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    19. *-rgt-identity N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \left(x \cdot \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    20. *-lowering-*.f64 67.9%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(x, \frac{5}{4}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 67.9%

    \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) + x \cdot 1.25}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 18: 68.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{e^{{x}^{2}}}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{{x}^{2}}\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left({x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 99.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. *-lowering-*.f64 76.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right)\right)\right), x\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 76.5%

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

13. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
14. Step-by-step derivation

    1. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    4. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left({x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    6. rgt-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    12. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    16. *-lowering-*.f64 67.9%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

15. Simplified 67.9%

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

Alternative 19: 68.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot x\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

   16. *-lowering-*.f64 76.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), 1\right)\right)\right)\right), x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 76.5%

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

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {x}^{3}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow3 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({x}^{2} \cdot x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

    9. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 67.9%

    \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)}}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 20: 2.3% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   2. associate-/r* N/A

      \[\leadsto \left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{x \cdot x} \cdot \left(\frac{1}{2} + \frac{1}{x \cdot x} \cdot \left(\frac{3}{4} + \frac{\frac{15}{8}}{x \cdot x}\right)\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{1}{x}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 2.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{/.f64}\left(\frac{15}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]

9. Simplified 2.3%

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

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 2.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}\right)\right) \]

12. Simplified 2.3%

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

    1. associate-/l/ N/A

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

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}{\color{blue}{x}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right), \color{blue}{x}\right) \]

    4. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}\right), x\right) \]

    5. pow-flip N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), x\right) \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right) \]

    7. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right) \]

    8. metadata-eval 2.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), x\right) \]

14. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))))