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

Percentage Accurate: 100.0% → 100.0%

Time: 1.3min

Alternatives: 18

Speedup: 1.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. *-lft-identity N/A

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

    2. associate-*l/ N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-/.f64 N/A

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

11. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{\left(\frac{1.875}{\left|x\right| \cdot \left(t\_0 \cdot t\_0\right)} + \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{t\_0 \cdot \left(x \cdot \left|x\right|\right)}\right)\right) \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (/
    (*
     (+
      (/ 1.875 (* (fabs x) (* t_0 t_0)))
      (+ (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x)) (/ 0.75 (* t_0 (* x (fabs x))))))
     (exp (* x x)))
    (sqrt PI))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return (((1.875 / (fabs(x) * (t_0 * t_0))) + ((((0.5 / (x * x)) + 1.0) / fabs(x)) + (0.75 / (t_0 * (x * fabs(x)))))) * exp((x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	return (((1.875 / (Math.abs(x) * (t_0 * t_0))) + ((((0.5 / (x * x)) + 1.0) / Math.abs(x)) + (0.75 / (t_0 * (x * Math.abs(x)))))) * Math.exp((x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	t_0 = x * (x * x)
	return (((1.875 / (math.fabs(x) * (t_0 * t_0))) + ((((0.5 / (x * x)) + 1.0) / math.fabs(x)) + (0.75 / (t_0 * (x * math.fabs(x)))))) * math.exp((x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(Float64(Float64(Float64(1.875 / Float64(abs(x) * Float64(t_0 * t_0))) + Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / abs(x)) + Float64(0.75 / Float64(t_0 * Float64(x * abs(x)))))) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = (((1.875 / (abs(x) * (t_0 * t_0))) + ((((0.5 / (x * x)) + 1.0) / abs(x)) + (0.75 / (t_0 * (x * abs(x)))))) * exp((x * x))) / sqrt(pi);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.875 / N[(N[Abs[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(t$95$0 * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{t\_0} + \frac{1.875}{\left(x \cdot x\right) \cdot t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* (* x x) (* x x)))))
   (*
    (/ (exp (* x x)) (sqrt PI))
    (+
     (/ (+ (/ 0.5 (* x x)) 1.0) (fabs x))
     (+ (/ 0.75 t_0) (/ 1.875 (* (* x x) t_0)))))))

C

double code(double x) {
	double t_0 = fabs(x) * ((x * x) * (x * x));
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((((0.5 / (x * x)) + 1.0) / fabs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * ((x * x) * (x * x));
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * ((((0.5 / (x * x)) + 1.0) / Math.abs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
}

Python

def code(x):
	t_0 = math.fabs(x) * ((x * x) * (x * x))
	return (math.exp((x * x)) / math.sqrt(math.pi)) * ((((0.5 / (x * x)) + 1.0) / math.fabs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(Float64(x * x) * Float64(x * x)))
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / abs(x)) + Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(Float64(x * x) * t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * ((x * x) * (x * x));
	tmp = (exp((x * x)) / sqrt(pi)) * ((((0.5 / (x * x)) + 1.0) / abs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / t$95$0), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 4: 99.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. unpow2 N/A

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

   3. fabs-sqr N/A

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

   4. unpow2 N/A

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

   5. fabs-mul N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fabs.f64 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 99.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 6: 99.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. unpow2 N/A

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

   7. fabs-sqr N/A

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

   8. unpow2 N/A

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

   9. fabs-mul N/A

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

   10. unpow2 N/A

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

   11. unpow3 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 99.5

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 7: 99.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around inf

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

    1. associate-*l/ N/A

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

    2. unpow2 N/A

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

    3. sqr-abs N/A

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

    4. unpow2 N/A

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

    5. associate-*r/ N/A

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

    6. times-frac N/A

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

    7. associate-*l/ N/A

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

    8. *-commutative N/A

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 8: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \frac{1}{\sqrt{\pi} \cdot \left(\left|x\right| \cdot e^{-x \cdot x}\right)} \]
13. Add Preprocessing

Alternative 9: 99.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 10: 99.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 11: 84.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.16666666666666666, 0.5)), 1.0), 1.0) / abs(x)))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 84.0%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|\color{blue}{x}\right|} \]
11. Add Preprocessing

Alternative 12: 81.0% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * fma(abs(x), fma(Float64(x * x), fma(Float64(x * x), 0.16666666666666666, 0.5), 1.0), Float64(1.0 / abs(x))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{\left|x\right|}}, \frac{1}{\left|x\right|}\right) \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right)}, \frac{1}{\left|x\right|}\right) \]
14. Add Preprocessing

Alternative 13: 81.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), Float64(abs(x) * fma(Float64(x * x), 0.16666666666666666, 0.5)), Float64(1.0 / abs(x))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.16666666666666666}{\left|x\right|}, \frac{0.5}{\left|x\right|}\right), \frac{1}{\left|x\right|}\right)}, \frac{1}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, {x}^{4} \cdot \left(\frac{1}{6} \cdot \frac{1}{\left|x\right|} + \color{blue}{\frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}}\right), \frac{1}{\left|x\right|}\right) \]
12. Step-by-step derivation

13. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{0.16666666666666666}, 0.5\right), \frac{1}{\left|x\right|}\right) \]
14. Add Preprocessing

Alternative 14: 81.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right), \frac{1}{\left|x\right|}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (fma (* x x) (* x (* x (* (fabs x) 0.16666666666666666))) (/ 1.0 (fabs x)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * fma((x * x), (x * (x * (fabs(x) * 0.16666666666666666))), (1.0 / fabs(x)));
}

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), Float64(x * Float64(x * Float64(abs(x) * 0.16666666666666666))), Float64(1.0 / abs(x))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.16666666666666666}{\left|x\right|}, \frac{0.5}{\left|x\right|}\right), \frac{1}{\left|x\right|}\right)}, \frac{1}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{6} \cdot \frac{{x}^{4}}{\color{blue}{\left|x\right|}}, \frac{1}{\left|x\right|}\right) \]
12. Step-by-step derivation

13. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \color{blue}{\left(0.16666666666666666 \cdot \left|x\right|\right)}\right), \frac{1}{\left|x\right|}\right) \]

14. Final simplification 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right), \frac{1}{\left|x\right|}\right) \]
15. Add Preprocessing

Alternative 15: 81.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 80.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.16666666666666666}{\left|x\right|}, \frac{0.5}{\left|x\right|}\right), \frac{1}{\left|x\right|}\right)}, \frac{1}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{6} \cdot \frac{{x}^{6}}{\color{blue}{\left|x\right|}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 80.5%

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

Alternative 16: 51.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right|} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(x, x, 1.0) / abs(x)))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 54.2%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\left|\color{blue}{x}\right|} \]
11. Add Preprocessing

Alternative 17: 5.4% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{\left|x\right|}}, \frac{1}{\left|x\right|}\right) \]
11. Step-by-step derivation

12. Applied rewrites 5.4%

    \[\leadsto \color{blue}{\frac{\left|x\right| + \frac{1}{\left|x\right|}}{\sqrt{\pi}}} \]
13. Add Preprocessing

Alternative 18: 5.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{\left|x\right|}}, \frac{1}{\left|x\right|}\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{{x}^{2}}{\left|x\right|} \]
12. Step-by-step derivation

13. Applied rewrites 5.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left|x\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024257 
(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)))))))