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

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

Alternatives: 9

Speedup: 8.3×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (*
    (pow PI -0.5)
    (fabs (* x (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)))))))
   (*
    0.047619047619047616
    (/ (* (fabs x) (* x (* x (* x (* x (* x x)))))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	} else {
		tmp = 0.047619047619047616 * ((fabs(x) * (x * (x * (x * (x * (x * x)))))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	} else {
		tmp = 0.047619047619047616 * ((Math.abs(x) * (x * (x * (x * (x * (x * x)))))) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))))
	else:
		tmp = 0.047619047619047616 * ((math.fabs(x) * (x * (x * (x * (x * (x * x)))))) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)))))));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(abs(x) * Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x)))))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	else
		tmp = 0.047619047619047616 * ((abs(x) * (x * (x * (x * (x * (x * x)))))) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 99.8%

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

   9. Simplified 99.8%

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

   if 0.0100000000000000002 < (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 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. cube-unmult N/A

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. *-commutative N/A

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

   7. Simplified 99.8%

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

      1. fabs-mul N/A

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

      2. fabs-sqr N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

      7. fabs-div N/A

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

      8. fabs-mul N/A

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

      9. fabs-fabs N/A

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

      10. metadata-eval N/A

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

      11. rem-sqrt-square N/A

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

      12. add-sqr-sqrt N/A

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

      13. un-div-inv N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. *-commutative N/A

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

   9. Applied egg-rr 99.9%

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

       1. associate-*l* N/A

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

       2. associate-*r* N/A

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

       3. pow3 N/A

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

       4. associate-*l/ N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-lft-identity N/A

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

       8. times-frac N/A

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

       9. metadata-eval N/A

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

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

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

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

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

   11. Applied egg-rr 100.0%

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

Alternative 3: 99.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (* (pow PI -0.5) (fabs (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (*
    0.047619047619047616
    (/ (* (fabs x) (* x (* x (* x (* x (* x x)))))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = 0.047619047619047616 * ((fabs(x) * (x * (x * (x * (x * (x * x)))))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	} else {
		tmp = 0.047619047619047616 * ((Math.abs(x) * (x * (x * (x * (x * (x * x)))))) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + (x * (x * 0.6666666666666666)))))
	else:
		tmp = 0.047619047619047616 * ((math.fabs(x) * (x * (x * (x * (x * (x * x)))))) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(abs(x) * Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x)))))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + (x * (x * 0.6666666666666666)))));
	else
		tmp = 0.047619047619047616 * ((abs(x) * (x * (x * (x * (x * (x * x)))))) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.5%

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

   9. Simplified 99.5%

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

   if 0.0100000000000000002 < (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 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. cube-unmult N/A

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. *-commutative N/A

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

   7. Simplified 99.8%

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

      1. fabs-mul N/A

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

      2. fabs-sqr N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

      7. fabs-div N/A

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

      8. fabs-mul N/A

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

      9. fabs-fabs N/A

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

      10. metadata-eval N/A

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

      11. rem-sqrt-square N/A

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

      12. add-sqr-sqrt N/A

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

      13. un-div-inv N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. *-commutative N/A

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

   9. Applied egg-rr 99.9%

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

       1. associate-*l* N/A

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

       2. associate-*r* N/A

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

       3. pow3 N/A

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

       4. associate-*l/ N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-lft-identity N/A

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

       8. times-frac N/A

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

       9. metadata-eval N/A

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

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

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

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

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

   11. Applied egg-rr 100.0%

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

Alternative 4: 89.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (* (pow PI -0.5) (fabs (* x (* 0.6666666666666666 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (0.6666666666666666 * (x * x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (0.6666666666666666 * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (0.6666666666666666 * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(0.6666666666666666 * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = (pi ^ -0.5) * abs((x * (0.6666666666666666 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   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. *-inverses N/A

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

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

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

      10. *-inverses N/A

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

      11. 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. un-div-inv N/A

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

      11. metadata-eval N/A

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

      12. sqrt-div N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      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 0.0100000000000000002 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 65.3%

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

   9. Simplified 65.3%

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

   10. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 65.3%

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

   12. Simplified 65.3%

       \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.8%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.8%

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

Alternative 6: 99.4% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.4%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. rem-sqrt-square N/A

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

   4. add-sqr-sqrt N/A

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

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

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

6. Applied egg-rr 99.4%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. distribute-lft-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. fabs-fabs N/A

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

   9. mul-fabs N/A

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 7: 89.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 87.6%

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

9. Simplified 87.6%

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

Alternative 8: 89.1% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.4%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. rem-sqrt-square N/A

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

   4. add-sqr-sqrt N/A

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

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

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

6. Applied egg-rr 99.4%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. distribute-lft-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. fabs-fabs N/A

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

   9. mul-fabs N/A

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 87.2%

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

11. Simplified 87.2%

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

12. Final simplification 87.2%

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

Alternative 9: 67.9% 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 99.8%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|} \]
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. *-inverses N/A

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

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

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

   10. *-inverses N/A

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

   11. PI-lowering-PI.f64 66.1%

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

7. Simplified 66.1%

   \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]
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. un-div-inv N/A

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

   11. metadata-eval N/A

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

   12. sqrt-div N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   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 66.1%

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

10. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2024161 
(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)))))))