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

Percentage Accurate: 100.0% → 100.0%

Time: 14.7s

Alternatives: 13

Speedup: 1.9×

• 99.3% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.75 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], 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 egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

   1. exp-prod N/A

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

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

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

   3. exp-lowering-exp.f64 100.0

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ e^{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\log \pi, -0.5, 0\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, 1 + \frac{0.5}{x \cdot x}, \frac{0.75}{x \cdot t\_0} + \frac{1.875}{x \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (*
    (exp (fma x x (fma (log PI) -0.5 0.0)))
    (fma
     (/ 1.0 (fabs x))
     (+ 1.0 (/ 0.5 (* x x)))
     (+ (/ 0.75 (* x t_0)) (/ 1.875 (* x (* (* x x) t_0))))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	return exp(fma(x, x, fma(log(((double) M_PI)), -0.5, 0.0))) * fma((1.0 / fabs(x)), (1.0 + (0.5 / (x * x))), ((0.75 / (x * t_0)) + (1.875 / (x * ((x * x) * t_0)))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	return Float64(exp(fma(x, x, fma(log(pi), -0.5, 0.0))) * fma(Float64(1.0 / abs(x)), Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(Float64(0.75 / Float64(x * t_0)) + Float64(1.875 / Float64(x * Float64(Float64(x * x) * t_0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Exp[N[(x * x + N[(N[Log[Pi], $MachinePrecision] * -0.5 + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

   3. pow1/2 N/A

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

   4. log-pow N/A

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

   5. +-rgt-identity N/A

      \[\leadsto \left(e^{\color{blue}{\left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right)} \cdot -1} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   6. sqr-abs N/A

      \[\leadsto \left(e^{\left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right) \cdot -1} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   7. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right) \cdot -1 + x \cdot x}} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   8. +-commutative N/A

      \[\leadsto e^{\color{blue}{x \cdot x + \left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right) \cdot -1}} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \color{blue}{e^{x \cdot x + \left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right) \cdot -1}} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

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

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, \left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right) \cdot -1\right)}} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   11. +-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. +-rgt-identity N/A

       \[\leadsto e^{\mathsf{fma}\left(x, x, -1 \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right) + 0\right)}\right)} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   14. distribute-rgt-in N/A

       \[\leadsto e^{\mathsf{fma}\left(x, x, \color{blue}{\left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right)\right) \cdot -1 + 0 \cdot -1}\right)} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto e^{\mathsf{fma}\left(x, x, \left(\frac{1}{2} \cdot \log \mathsf{PI}\left(\right)\right) \cdot -1 + \color{blue}{0}\right)} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{1}{2}}{x \cdot x} + 1, \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{\frac{15}{8}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\log \pi, -0.5, 0\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{0.5}{x \cdot x} + 1, \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

9. Final simplification 100.0%

   \[\leadsto e^{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\log \pi, -0.5, 0\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, 1 + \frac{0.5}{x \cdot x}, \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * 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 egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-*l/ N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

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

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

   8. sqr-abs N/A

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

   9. *-lft-identity N/A

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

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

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

   11. *-lowering-*.f64 100.0

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.75 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * 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 egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.75 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $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 egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 6: 99.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, x \cdot x, 0\right), 0\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (/ 0.75 (fma x (fma (* x x) (* x x) 0.0) 0.0)))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + (0.75 / fma(x, fma((x * x), (x * x), 0.0), 0.0)));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(0.75 / fma(x, fma(Float64(x * x), Float64(x * x), 0.0), 0.0))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

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

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

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

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

   3. *-commutative N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. +-commutative N/A

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

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

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

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

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

   13. fabs-lowering-fabs.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-commutative N/A

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

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

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

   19. fabs-lowering-fabs.f64 N/A

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

   20. +-rgt-identity N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{1}{2}}{\left|x\right| \cdot \left(x \cdot x\right)} + \left(\frac{1}{\left|x\right|} + \frac{\frac{3}{4}}{\left|x\right| \cdot \color{blue}{\left({x}^{4} + 0\right)}}\right)\right) \]

7. Simplified 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, x \cdot x, 0\right), 0\right)}\right) \]
11. Add Preprocessing

Alternative 7: 99.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $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 egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

   8. associate-*r/ N/A

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

   9. times-frac N/A

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

   10. distribute-lft1-in N/A

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 8: 99.4% 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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 99.4%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. frac-times N/A

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

   4. sqr-abs N/A

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

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

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

   6. sqr-abs N/A

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

   7. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   13. fabs-lowering-fabs.f64 99.4

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

9. Applied egg-rr 99.4%

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

Alternative 9: 84.1% accurate, 6.4× speedup

Math

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

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(x, Float64(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[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 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

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 83.3

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

10. Simplified 83.3%

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

Alternative 10: 80.8% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[Abs[x], $MachinePrecision] + N[(0.5 / N[Abs[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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 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

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

10. Simplified 80.8%

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

11. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. associate-*r* N/A

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

    9. associate-*r* N/A

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

    10. distribute-rgt-out N/A

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

13. Simplified 80.8%

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

Alternative 11: 80.8% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.16666666666666666), $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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 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

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

10. Simplified 80.8%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{6}} \]

    2. associate-*l* N/A

       \[\leadsto \color{blue}{\left({x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)} \cdot \frac{1}{6} \]

    3. *-commutative N/A

       \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right) \cdot \frac{1}{6} \]

    4. associate-*r* N/A

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \frac{1}{6}\right)} \]

    5. *-commutative N/A

       \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

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

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

    7. metadata-eval N/A

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

    8. pow-sqr N/A

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

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

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

    10. unpow2 N/A

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

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

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

    12. unpow2 N/A

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

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

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

    14. *-commutative N/A

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

    15. associate-*l* N/A

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

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

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

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

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

    18. /-lowering-/.f64 N/A

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

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

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

    20. *-commutative N/A

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

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

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

    22. fabs-lowering-fabs.f64 80.8

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

13. Simplified 80.8%

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

14. Final simplification 80.8%

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

Alternative 12: 67.9% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.5 * 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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. associate-/l* N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. fabs-mul N/A

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

   15. *-rgt-identity N/A

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

10. Simplified 67.6%

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

11. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

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

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

    5. unpow2 N/A

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

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

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

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

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

    8. fabs-lowering-fabs.f64 67.6

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

13. Simplified 67.6%

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

Alternative 13: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / 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 egg-rr 100.0%

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

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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 99.4

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

7. Simplified 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

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

   7. fabs-lowering-fabs.f64 2.4

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

10. Simplified 2.4%

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

    1. div-inv N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. frac-times N/A

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

    5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

    10. fabs-lowering-fabs.f64 2.4

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

12. Applied egg-rr 2.4%

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

Reproduce

herbie shell --seed 2024199 
(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)))))))