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

Percentage Accurate: 99.8% → 99.8%

Time: 13.4s

Alternatives: 7

Speedup: 8.3×

• 29.3% 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, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot {x}^{4}\right)\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+
       (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0))
       (* (/ 1.0 5.0) (* (fabs x) (* (fabs x) t_0))))
      (* (/ 1.0 21.0) (* (fabs x) (* (fabs x) (* (fabs x) (pow x 4.0))))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * (fabs(x) * (fabs(x) * t_0)))) + ((1.0 / 21.0) * (fabs(x) * (fabs(x) * (fabs(x) * pow(x, 4.0))))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * 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) * (Math.abs(x) * (Math.abs(x) * t_0)))) + ((1.0 / 21.0) * (Math.abs(x) * (Math.abs(x) * (Math.abs(x) * Math.pow(x, 4.0))))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * 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) * (math.fabs(x) * (math.fabs(x) * t_0)))) + ((1.0 / 21.0) * (math.fabs(x) * (math.fabs(x) * (math.fabs(x) * math.pow(x, 4.0))))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * 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) * Float64(abs(x) * Float64(abs(x) * t_0)))) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * Float64(abs(x) * (x ^ 4.0))))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * (abs(x) * (abs(x) * t_0)))) + ((1.0 / 21.0) * (abs(x) * (abs(x) * (abs(x) * (x ^ 4.0))))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * 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] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Power[x, 4.0], $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
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. pow2 N/A

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

   3. sqr-abs N/A

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

   4. unpow-prod-down N/A

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

   5. pow-prod-up N/A

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

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

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

   7. metadata-eval 99.9%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{fabs.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, 3\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, 5\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, 21\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right), \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{{x}^{4}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * 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) * (Math.abs(x) * (Math.abs(x) * t_0)))) + ((1.0 / 21.0) * (Math.abs(x) * (Math.abs(x) * (Math.abs(x) * ((x * x) * (x * x)))))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * 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) * (math.fabs(x) * (math.fabs(x) * t_0)))) + ((1.0 / 21.0) * (math.fabs(x) * (math.fabs(x) * (math.fabs(x) * ((x * x) * (x * x)))))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * 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) * Float64(abs(x) * Float64(abs(x) * t_0)))) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * Float64(abs(x) * Float64(Float64(x * x) * Float64(x * x)))))))))
end

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * 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] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $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
3. Step-by-step derivation

   1. associate-*l* N/A

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

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

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

   3. sqr-abs N/A

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

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

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

   5. sqr-abs N/A

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

   6. *-lowering-*.f64 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 94.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\frac{\left(x \cdot x\right) \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.2\right)\right)\right)\right)}{x}\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-9)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (/
    (fabs
     (/
      (* (* x x) (+ 2.0 (* x (* x (+ 0.6666666666666666 (* x (* x 0.2)))))))
      x))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-9) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((((x * x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.2))))))) / x)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-9) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((((x * x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.2))))))) / x)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-9:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((((x * x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.2))))))) / x)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-9)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(abs(Float64(Float64(Float64(x * x) * Float64(2.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(x * Float64(x * 0.2))))))) / x)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-9)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = abs((((x * x) * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.2))))))) / x)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e-9], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(x * N[(x * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

      1. associate-*l/ N/A

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

      2. fabs-div N/A

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

      3. *-lft-identity N/A

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

      4. fabs-mul N/A

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

      5. fabs-fabs N/A

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

      6. metadata-eval N/A

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

      7. rem-sqrt-square N/A

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

      8. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. div-inv N/A

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

       2. metadata-eval N/A

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

       3. sqrt-div N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. fabs-lowering-fabs.f64 99.9%

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

   11. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-9 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied egg-rr 13.0%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{\frac{\left|x\right| \cdot \left(\left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right) - \left(x \cdot x\right) \cdot \left(0.2 \cdot \left(x \cdot x\right) + 0.047619047619047616 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\left(x \cdot x\right) \cdot \left(\left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) - \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.2 \cdot \left(x \cdot x\right) + 0.047619047619047616 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.2 \cdot \left(x \cdot x\right) + 0.047619047619047616 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}}}\right| \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 76.9%

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

   9. Simplified 76.9%

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

       1. fabs-div N/A

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

       2. rem-sqrt-square N/A

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

       3. add-sqr-sqrt N/A

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

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

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

   11. Applied egg-rr 80.8%

       \[\leadsto \color{blue}{\frac{\left|\frac{\left(x \cdot x\right) \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.2\right)\right)\right)\right)}{x}\right|}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\frac{\left(x \cdot x\right) \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.2\right)\right)\right)\right)}{x}\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-9)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (*
    (* x x)
    (*
     (/ (sqrt (/ 1.0 PI)) (fabs x))
     (+ 2.0 (* (* x x) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-9) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (x * x) * ((sqrt((1.0 / ((double) M_PI))) / fabs(x)) * (2.0 + ((x * x) * 0.6666666666666666)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-9) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (x * x) * ((Math.sqrt((1.0 / Math.PI)) / Math.abs(x)) * (2.0 + ((x * x) * 0.6666666666666666)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-9:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (x * x) * ((math.sqrt((1.0 / math.pi)) / math.fabs(x)) * (2.0 + ((x * x) * 0.6666666666666666)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-9)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(sqrt(Float64(1.0 / pi)) / abs(x)) * Float64(2.0 + Float64(Float64(x * x) * 0.6666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-9)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = (x * x) * ((sqrt((1.0 / pi)) / abs(x)) * (2.0 + ((x * x) * 0.6666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e-9], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

      1. associate-*l/ N/A

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

      2. fabs-div N/A

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

      3. *-lft-identity N/A

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

      4. fabs-mul N/A

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

      5. fabs-fabs N/A

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

      6. metadata-eval N/A

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

      7. rem-sqrt-square N/A

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

      8. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. div-inv N/A

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

       2. metadata-eval N/A

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

       3. sqrt-div N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. fabs-lowering-fabs.f64 99.9%

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

   11. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-9 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied egg-rr 13.0%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right) \cdot {x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left(\left|x\right| \cdot \left(\frac{1}{6} - \frac{1}{3}\right)\right) \cdot {x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left|x\right| \cdot \left(\left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{6} - \frac{1}{3}\right), \left({x}^{2}\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 3.4%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   7. Simplified 3.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\color{blue}{\frac{\left|x\right| \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{x \cdot x}}}\right| \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. fabs-div N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. fabs-sqr N/A

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

      7. sqr-pow N/A

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

      8. inv-pow N/A

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

      9. rem-sqrt-square N/A

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

      10. add-sqr-sqrt N/A

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

   9. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \frac{\left|x\right|}{x \cdot x}}} \]

   10. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l/ N/A

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

       7. associate-/l* N/A

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

       8. *-rgt-identity N/A

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

       9. associate-*r/ N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

   12. Simplified 67.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\left|x\right| \cdot 0.5}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e+25)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (/ (pow PI -0.5) (/ (* (fabs x) 0.5) (* x x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e+25) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(((double) M_PI), -0.5) / ((fabs(x) * 0.5) / (x * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e+25) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(Math.PI, -0.5) / ((Math.abs(x) * 0.5) / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e+25:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(math.pi, -0.5) / ((math.fabs(x) * 0.5) / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e+25)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((pi ^ -0.5) / Float64(Float64(abs(x) * 0.5) / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e+25)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = (pi ^ -0.5) / ((abs(x) * 0.5) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+25], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] / N[(N[(N[Abs[x], $MachinePrecision] * 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e25

   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
   3. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 93.1%

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

      1. associate-*l/ N/A

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

      2. fabs-div N/A

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

      3. *-lft-identity N/A

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

      4. fabs-mul N/A

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

      5. fabs-fabs N/A

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

      6. metadata-eval N/A

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

      7. rem-sqrt-square N/A

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

      8. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 92.4%

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

   9. Applied egg-rr 92.4%

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

       1. div-inv N/A

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

       2. metadata-eval N/A

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

       3. sqrt-div N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. fabs-lowering-fabs.f64 93.1%

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

   11. Applied egg-rr 93.1%

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

   if 5.00000000000000024e25 < (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 0.7%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left(\frac{1}{6} \cdot \left|x\right| - \frac{1}{3} \cdot \left|x\right|\right) \cdot {x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left(\left|x\right| \cdot \left(\frac{1}{6} - \frac{1}{3}\right)\right) \cdot {x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \frac{1}{2} + \left|x\right| \cdot \left(\left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| \cdot \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{1}{2} + \left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{6} - \frac{1}{3}\right) \cdot {x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{6} - \frac{1}{3}\right), \left({x}^{2}\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   7. Simplified 1.0%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\color{blue}{\frac{\left|x\right| \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{x \cdot x}}}\right| \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. fabs-div N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. fabs-sqr N/A

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

      7. sqr-pow N/A

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

      8. inv-pow N/A

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

      9. rem-sqrt-square N/A

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

      10. add-sqr-sqrt N/A

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

   9. Applied egg-rr 0.4%

      \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{\left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \frac{\left|x\right|}{x \cdot x}}} \]

   10. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \frac{1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 60.2%

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

   12. Simplified 60.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\pi}^{-0.5}}{\frac{\left|x\right| \cdot 0.5}{x \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 8.3× speedup

Math

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

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 abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(x * Float64(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[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(x * N[(x * N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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
3. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. associate-*r* N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left|\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right|} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{fabs.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left|x\right| \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(2 + \left(\frac{2}{3} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right|} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 7: 67.4% 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%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. associate-*r* N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 65.2%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. *-lft-identity N/A

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

   4. fabs-mul N/A

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

   5. fabs-fabs N/A

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

   6. metadata-eval N/A

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

   7. rem-sqrt-square N/A

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

   8. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

   13. PI-lowering-PI.f64 64.7%

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

9. Applied egg-rr 64.7%

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

    1. div-inv N/A

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

    2. metadata-eval N/A

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

    3. sqrt-div N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

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

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

    7. sqrt-div N/A

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

    8. metadata-eval N/A

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

    9. div-inv N/A

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

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

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

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

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

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

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

    13. fabs-lowering-fabs.f64 65.2%

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

11. Applied egg-rr 65.2%

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

12. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024191 
(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)))))))