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

Percentage Accurate: 99.9% → 99.9%

Time: 11.4s

Alternatives: 12

Speedup: 1.5×

• 28.6% 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: 99.9% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (* x (* x x)))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (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) (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs((x * (x * x)));
	double t_1 = fabs(x) * (fabs(x) * t_0);
	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) * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs((x * (x * x)));
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	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) * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs((x * (x * x)))
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	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) * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = abs(Float64(x * Float64(x * x)))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	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(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs((x * (x * x)));
	t_1 = abs(x) * (abs(x) * t_0);
	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) * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 88.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1.0)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (fabs (* (sqrt (/ 1.0 PI)) (* 0.6666666666666666 (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.0) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (0.6666666666666666 * (x * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1.0) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (0.6666666666666666 * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1.0:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (0.6666666666666666 * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.0)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.6666666666666666 * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1.0)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = abs((sqrt((1.0 / pi)) * (0.6666666666666666 * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   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 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 98.5%

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

   7. Simplified 98.5%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

      4. div-fabs N/A

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

      5. fabs-mul N/A

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

      6. fabs-fabs N/A

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

      7. metadata-eval N/A

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

      8. rem-sqrt-square N/A

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

      9. add-sqr-sqrt N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. un-div-inv N/A

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

      13. metadata-eval N/A

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

      14. sqrt-div N/A

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

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

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

   9. Applied egg-rr 98.5%

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

   if 1 < (fabs.f64 x)

   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 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 71.2%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. fabs-mul N/A

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

      3. fabs-fabs N/A

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

      4. mul-fabs N/A

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

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

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

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

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

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left|\frac{2 + 0.6666666666666666 \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot x\right|} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 71.2%

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

   11. Applied egg-rr 71.2%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{2}{3} \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

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

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

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

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. unpow3 N/A

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

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

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

       17. cube-mult N/A

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

       18. unpow2 N/A

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

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

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

       20. unpow2 N/A

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

       21. *-lowering-*.f64 72.2%

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

   14. Simplified 72.2%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1.0)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (fabs (* x (/ (* x (* x 0.6666666666666666)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.0) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((x * ((x * (x * 0.6666666666666666)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1.0) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((x * ((x * (x * 0.6666666666666666)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1.0:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((x * ((x * (x * 0.6666666666666666)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.0)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = abs(Float64(x * Float64(Float64(x * Float64(x * 0.6666666666666666)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1.0)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = abs((x * ((x * (x * 0.6666666666666666)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   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 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 98.5%

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

   7. Simplified 98.5%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

      4. div-fabs N/A

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

      5. fabs-mul N/A

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

      6. fabs-fabs N/A

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

      7. metadata-eval N/A

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

      8. rem-sqrt-square N/A

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

      9. add-sqr-sqrt N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. un-div-inv N/A

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

      13. metadata-eval N/A

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

      14. sqrt-div N/A

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

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

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

   9. Applied egg-rr 98.5%

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

   if 1 < (fabs.f64 x)

   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 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 71.2%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. fabs-mul N/A

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

      3. fabs-fabs N/A

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

      4. mul-fabs N/A

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

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

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

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

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

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left|\frac{2 + 0.6666666666666666 \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot x\right|} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 71.2%

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

   11. Applied egg-rr 71.2%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 71.2%

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

   14. Simplified 71.2%

       \[\leadsto \left|\frac{\color{blue}{x \cdot \left(x \cdot 0.6666666666666666\right)}}{\sqrt{\pi}} \cdot x\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 8.3× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

Alternative 5: 99.4% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow1/2 N/A

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

   4. fabs-mul N/A

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

   5. associate-*r* N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. add-sqr-sqrt N/A

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

   9. rem-sqrt-square N/A

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

   10. div-fabs N/A

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

   11. mul-fabs N/A

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

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

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 99.4%

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

Alternative 6: 99.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * (0.6666666666666666 + (0.047619047619047616 * (x * (x * (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(0.047619047619047616 * Float64(x * Float64(x * Float64(x * x)))))))))))
end

MATLAB

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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf

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

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{4}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. metadata-eval N/A

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

    3. pow-plus N/A

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

    4. unpow3 N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. unpow2 N/A

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

    8. unpow2 N/A

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

    9. associate-*l* N/A

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

    10. unpow2 N/A

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

    11. cube-mult N/A

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

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

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

    13. cube-mult N/A

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

    14. unpow2 N/A

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

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

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

    16. unpow2 N/A

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

    17. *-lowering-*.f64 98.9%

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

11. Simplified 98.9%

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

Alternative 7: 98.6% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow1/2 N/A

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

   4. fabs-mul N/A

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

   5. associate-*r* N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. add-sqr-sqrt N/A

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

   9. rem-sqrt-square N/A

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

   10. div-fabs N/A

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

   11. mul-fabs N/A

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

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

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 99.4%

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

11. Taylor expanded in x around inf

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

    1. unpow3 N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

    6. *-commutative N/A

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

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

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

    8. unpow2 N/A

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

    9. *-lowering-*.f64 98.5%

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

13. Simplified 98.5%

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

Alternative 8: 93.5% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

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)))))));
}

Python

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

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) * 0.2))))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * 0.2)))))));
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] * 0.2), $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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]
10. 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 95.4%

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

11. Simplified 95.4%

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

Alternative 9: 93.1% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. inv-pow N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(\left|\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\mathsf{PI}\left(\right)}^{-1}}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{-1}{2}\right)}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(\left|\left|x\right| \cdot \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right| \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\color{blue}{\left|x\right|} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

   11. fabs-mul N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|\left|x\right|\right| \cdot \color{blue}{\left|2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right|}\right)\right) \]

   12. fabs-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x\right| \cdot \left|\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}\right|\right)\right) \]

   13. mul-fabs N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\left|x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{fabs.f64}\left(\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow1/2 N/A

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

   4. fabs-mul N/A

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

   5. associate-*r* N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. add-sqr-sqrt N/A

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

   9. rem-sqrt-square N/A

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

   10. div-fabs N/A

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

   11. mul-fabs N/A

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

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

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 + x \cdot \left(x \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 99.4%

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

11. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 94.9%

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

13. Simplified 94.9%

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

Alternative 10: 89.2% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

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

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

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

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

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

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

7. Simplified 90.6%

   \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 90.6%

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

10. Final simplification 90.6%

    \[\leadsto \left|x \cdot \frac{2 + 0.6666666666666666 \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 11: 88.7% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

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

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

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

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

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

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

7. Simplified 90.6%

   \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 90.6%

   \[\leadsto \color{blue}{\left|\frac{2 + 0.6666666666666666 \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot x\right|} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

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

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

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

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

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

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

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

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

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

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

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

    10. *-lowering-*.f64 90.1%

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

11. Applied egg-rr 90.1%

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

Alternative 12: 67.8% 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

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f64 69.5%

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

7. Simplified 69.5%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

   4. div-fabs N/A

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

   5. fabs-mul N/A

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

   6. fabs-fabs N/A

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

   7. metadata-eval N/A

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

   8. rem-sqrt-square N/A

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

   9. add-sqr-sqrt N/A

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

   10. *-commutative N/A

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

   11. associate-*l/ N/A

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

   12. un-div-inv N/A

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

   13. metadata-eval N/A

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

   14. sqrt-div N/A

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

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

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

9. Applied egg-rr 69.5%

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

10. Final simplification 69.5%

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

Reproduce

herbie shell --seed 2024164 
(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)))))))