Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 11

Speedup: 8.3×

• 29.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (*
    (fabs x)
    (+
     2.0
     (*
      x
      (*
       x
       (+
        0.6666666666666666
        (* x (* x (+ 0.2 (* 0.047619047619047616 (* x x)))))))))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (fabs(x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.2 + (0.047619047619047616 * (x * x))))))))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (Math.abs(x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.2 + (0.047619047619047616 * (x * x))))))))))));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (math.fabs(x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.2 + (0.047619047619047616 * (x * x))))))))))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * Float64(2.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(x * Float64(x * Float64(0.2 + Float64(0.047619047619047616 * Float64(x * x))))))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (abs(x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.2 + (0.047619047619047616 * (x * x))))))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(x * N[(x * N[(0.2 + N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + 0.2 \cdot \left(x \cdot x\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{\left|x\right| \cdot 0.047619047619047616}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (*
    (pow PI -0.5)
    (fabs (* x (+ 2.0 (* x (* x (+ 0.6666666666666666 (* 0.2 (* x x)))))))))
   (/
    (* x (* (* x x) (* x (* x x))))
    (/ (sqrt PI) (* (fabs x) 0.047619047619047616)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * (0.6666666666666666 + (0.2 * (x * x))))))));
	} else {
		tmp = (x * ((x * x) * (x * (x * x)))) / (sqrt(((double) M_PI)) / (fabs(x) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * (0.6666666666666666 + (0.2 * (x * x))))))));
	} else {
		tmp = (x * ((x * x) * (x * (x * x)))) / (Math.sqrt(Math.PI) / (Math.abs(x) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * (0.6666666666666666 + (0.2 * (x * x))))))))
	else:
		tmp = (x * ((x * x) * (x * (x * x)))) / (math.sqrt(math.pi) / (math.fabs(x) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(0.2 * Float64(x * x)))))))));
	else
		tmp = Float64(Float64(x * Float64(Float64(x * x) * Float64(x * Float64(x * x)))) / Float64(sqrt(pi) / Float64(abs(x) * 0.047619047619047616)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * (0.6666666666666666 + (0.2 * (x * x))))))));
	else
		tmp = (x * ((x * x) * (x * (x * x)))) / (sqrt(pi) / (abs(x) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 99.6%

       \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)\right)}\right)\right| \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.0%

      \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left|\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|x \cdot x\right| \cdot \color{blue}{\left|\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right|} \]

      3. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left|\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)}\right| \]

      4. fabs-mul N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right| \cdot \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right| \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|\right) \]

      6. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right|\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      11. un-div-inv N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      12. fabs-div N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left|\frac{1}{21} \cdot \left|x\right|\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}}\right) \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{0.047619047619047616 \cdot \left|x\right|}{\sqrt{\pi}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

       3. clear-num N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \color{blue}{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\frac{1}{21} \cdot \left|x\right|\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\color{blue}{\frac{1}{21}} \cdot \left|x\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\frac{1}{21}, \color{blue}{\left(\left|x\right|\right)}\right)\right)\right) \]

       16. fabs-lowering-fabs.f64 99.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right) \]

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot \left|x\right|}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + 0.2 \cdot \left(x \cdot x\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{\left|x\right| \cdot 0.047619047619047616}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{\left|x\right| \cdot 0.047619047619047616}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (* (pow PI -0.5) (fabs (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (/
    (* x (* (* x x) (* x (* x x))))
    (/ (sqrt PI) (* (fabs x) 0.047619047619047616)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = (x * ((x * x) * (x * (x * x)))) / (sqrt(((double) M_PI)) / (fabs(x) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = (x * ((x * x) * (x * (x * x)))) / (Math.sqrt(Math.PI) / (Math.abs(x) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * 0.6666666666666666)))))
	else:
		tmp = (x * ((x * x) * (x * (x * x)))) / (math.sqrt(math.pi) / (math.fabs(x) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = Float64(Float64(x * Float64(Float64(x * x) * Float64(x * Float64(x * x)))) / Float64(sqrt(pi) / Float64(abs(x) * 0.047619047619047616)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	else
		tmp = (x * ((x * x) * (x * (x * x)))) / (sqrt(pi) / (abs(x) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 99.6%

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

   12. Simplified 99.6%

       \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot 0.6666666666666666\right)}\right)\right| \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.0%

      \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left|\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|x \cdot x\right| \cdot \color{blue}{\left|\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right|} \]

      3. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left|\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)}\right| \]

      4. fabs-mul N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right| \cdot \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right| \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|\right) \]

      6. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right|\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      11. un-div-inv N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      12. fabs-div N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left|\frac{1}{21} \cdot \left|x\right|\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}}\right) \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{0.047619047619047616 \cdot \left|x\right|}{\sqrt{\pi}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}} \]

       3. clear-num N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \color{blue}{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{21} \cdot \left|x\right|}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\frac{1}{21} \cdot \left|x\right|\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\color{blue}{\frac{1}{21}} \cdot \left|x\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\frac{1}{21}, \color{blue}{\left(\left|x\right|\right)}\right)\right)\right) \]

       16. fabs-lowering-fabs.f64 99.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right) \]

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot \left|x\right|}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\frac{\sqrt{\pi}}{\left|x\right| \cdot 0.047619047619047616}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\left|x\right| \cdot 0.047619047619047616}{\sqrt{\pi}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (* (pow PI -0.5) (fabs (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (*
    (* x (/ (* (fabs x) 0.047619047619047616) (sqrt PI)))
    (* x (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = (x * ((fabs(x) * 0.047619047619047616) / sqrt(((double) M_PI)))) * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = (x * ((Math.abs(x) * 0.047619047619047616) / Math.sqrt(Math.PI))) * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * 0.6666666666666666)))))
	else:
		tmp = (x * ((math.fabs(x) * 0.047619047619047616) / math.sqrt(math.pi))) * (x * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = Float64(Float64(x * Float64(Float64(abs(x) * 0.047619047619047616) / sqrt(pi))) * Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	else
		tmp = (x * ((abs(x) * 0.047619047619047616) / sqrt(pi))) * (x * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 99.6%

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

   12. Simplified 99.6%

       \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot 0.6666666666666666\right)}\right)\right| \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.0%

      \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left|\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|x \cdot x\right| \cdot \color{blue}{\left|\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right|} \]

      3. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left|\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)}\right| \]

      4. fabs-mul N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right| \cdot \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right| \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|\right) \]

      6. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right|\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      11. un-div-inv N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      12. fabs-div N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left|\frac{1}{21} \cdot \left|x\right|\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}}\right) \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{0.047619047619047616 \cdot \left|x\right|}{\sqrt{\pi}} \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\left|x\right| \cdot 0.047619047619047616}{\sqrt{\pi}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\frac{\left|x\right| \cdot 0.047619047619047616}{\sqrt{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (* (pow PI -0.5) (fabs (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (*
    x
    (*
     x
     (*
      (/ (* (fabs x) 0.047619047619047616) (sqrt PI))
      (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = x * (x * (((fabs(x) * 0.047619047619047616) / sqrt(((double) M_PI))) * (x * (x * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = x * (x * (((Math.abs(x) * 0.047619047619047616) / Math.sqrt(Math.PI)) * (x * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * 0.6666666666666666)))))
	else:
		tmp = x * (x * (((math.fabs(x) * 0.047619047619047616) / math.sqrt(math.pi)) * (x * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(Float64(abs(x) * 0.047619047619047616) / sqrt(pi)) * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	else
		tmp = x * (x * (((abs(x) * 0.047619047619047616) / sqrt(pi)) * (x * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 99.6%

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

   12. Simplified 99.6%

       \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot 0.6666666666666666\right)}\right)\right| \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{21} \cdot \left({x}^{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left({x}^{6} \cdot \frac{1}{21}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{21} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)\right)\right) \]

   7. Simplified 99.0%

      \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left|\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right)\right| \]

      2. fabs-mul N/A

         \[\leadsto \left|x \cdot x\right| \cdot \color{blue}{\left|\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)\right|} \]

      3. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left|\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right)}\right| \]

      4. fabs-mul N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right| \cdot \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left|\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right| \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right|\right) \]

      6. fabs-sqr N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}\right|\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right|\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      11. un-div-inv N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\frac{\frac{1}{21} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right) \]

      12. fabs-div N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left|\frac{1}{21} \cdot \left|x\right|\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}}\right) \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.047619047619047616 \cdot \left|x\right|}{\sqrt{\pi}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\frac{\left|x\right| \cdot 0.047619047619047616}{\sqrt{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow PI -0.5)
  (fabs
   (*
    x
    (+
     2.0
     (*
      x
      (*
       x
       (+
        0.6666666666666666
        (* (* x x) (+ 0.2 (* x (* x 0.047619047619047616))))))))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616))))))))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616))))))))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616))))))))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(Float64(x * x) * Float64(0.2 + Float64(x * Float64(x * 0.047619047619047616)))))))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616))))))))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.2 + N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]
10. Add Preprocessing

Alternative 7: 99.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow PI -0.5)
  (fabs
   (*
    x
    (+
     2.0
     (*
      x
      (*
       x
       (+
        0.6666666666666666
        (* x (* x (* 0.047619047619047616 (* x x))))))))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.047619047619047616 * (x * x))))))))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.047619047619047616 * (x * x))))))))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.047619047619047616 * (x * x))))))))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(x * Float64(x * Float64(0.047619047619047616 * Float64(x * x)))))))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * (0.047619047619047616 * (x * x))))))))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(x * N[(x * N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{1}{21} \cdot {x}^{4}\right)}\right)\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{21} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\frac{1}{21} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 99.4%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 99.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right)\right)\right)\right| \]

13. Final simplification 99.4%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right| \]
14. Add Preprocessing

Alternative 8: 99.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow PI -0.5)
  (fabs
   (* x (+ 2.0 (* x (* (* x x) (* x (* 0.047619047619047616 (* x x))))))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * ((x * x) * (x * (0.047619047619047616 * (x * x))))))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * ((x * x) * (x * (0.047619047619047616 * (x * x))))))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * ((x * x) * (x * (0.047619047619047616 * (x * x))))))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(Float64(x * x) * Float64(x * Float64(0.047619047619047616 * Float64(x * x)))))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * ((x * x) * (x * (0.047619047619047616 * (x * x))))))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{5}\right)}\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{\left(4 + 1\right)}\right)\right)\right)\right)\right)\right) \]

    2. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{4} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{4}\right) \cdot x\right)\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4}\right)\right)\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right)\right) \]

    6. pow-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    17. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

    19. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

    20. *-lowering-*.f64 99.2%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 99.2%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}\right)\right| \]

13. Final simplification 99.2%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
14. Add Preprocessing

Alternative 9: 98.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x \cdot \left(-2 + x \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot -0.047619047619047616\right)\right)\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (* x (+ -2.0 (* x (* (* (* x x) (* x (* x x))) -0.047619047619047616)))))
  (sqrt PI)))

C

double code(double x) {
	return fabs((x * (-2.0 + (x * (((x * x) * (x * (x * x))) * -0.047619047619047616))))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return Math.abs((x * (-2.0 + (x * (((x * x) * (x * (x * x))) * -0.047619047619047616))))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs((x * (-2.0 + (x * (((x * x) * (x * (x * x))) * -0.047619047619047616))))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(abs(Float64(x * Float64(-2.0 + Float64(x * Float64(Float64(Float64(x * x) * Float64(x * Float64(x * x))) * -0.047619047619047616))))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (-2.0 + (x * (((x * x) * (x * (x * x))) * -0.047619047619047616))))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[Abs[N[(x * N[(-2.0 + N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{5}\right)}\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{\left(4 + 1\right)}\right)\right)\right)\right)\right)\right) \]

    2. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{4} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{4}\right) \cdot x\right)\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4}\right)\right)\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right)\right) \]

    6. pow-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    11. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

    17. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

    19. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

    20. *-lowering-*.f64 99.2%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 99.2%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)}\right)\right| \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right| \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}} \]

    2. metadata-eval N/A

       \[\leadsto \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right| \cdot {\mathsf{PI}\left(\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

    3. pow-flip N/A

       \[\leadsto \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right| \cdot \frac{1}{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}} \]

    4. pow1/2 N/A

       \[\leadsto \left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right| \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \]

    5. un-div-inv N/A

       \[\leadsto \frac{\left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\left|x \cdot \left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right|\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

14. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{\left|x \cdot \left(-2 + x \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot -0.047619047619047616\right)\right)\right|}{\sqrt{\pi}}} \]
15. Add Preprocessing

Alternative 10: 89.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow PI -0.5) (fabs (* x (+ 2.0 (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * 0.6666666666666666)))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * 0.6666666666666666)))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|} \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)\right|} \]

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 88.5%

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

12. Simplified 88.5%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot 0.6666666666666666\right)}\right)\right| \]
13. Add Preprocessing

Alternative 11: 68.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \frac{2}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

double code(double x) {
	return fabs(x) * (2.0 / sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return math.fabs(x) * (2.0 / math.sqrt(math.pi))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-inverses N/A

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

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

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

   8. *-inverses N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. fabs-lowering-fabs.f64 65.7%

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

7. Simplified 65.7%

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

   1. associate-*r* N/A

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

   2. fabs-mul N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. fabs-div N/A

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

   8. fabs-fabs N/A

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

   9. rem-sqrt-square N/A

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

   10. add-sqr-sqrt N/A

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

   11. clear-num N/A

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

   12. metadata-eval N/A

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

   13. associate-*l/ N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 N/A

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

9. Applied egg-rr 65.0%

   \[\leadsto \color{blue}{\frac{2}{\left|\frac{\sqrt{\pi}}{x}\right|}} \]
10. Step-by-step derivation

    1. fabs-div N/A

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

    2. rem-sqrt-square N/A

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

    3. add-sqr-sqrt N/A

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

    4. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

    9. fabs-lowering-fabs.f64 65.7%

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

11. Applied egg-rr 65.7%

    \[\leadsto \color{blue}{\frac{2}{\sqrt{\pi}} \cdot \left|x\right|} \]

12. Final simplification 65.7%

    \[\leadsto \left|x\right| \cdot \frac{2}{\sqrt{\pi}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))